\label{t:main}
  Let $q\ge4$ be a prime power and $G$ be almost simple with socle
  $S=\PSL_2(q)$, that is $S\le G\le\aut(S)$. Suppose that $G$ acts
  primitively on $\Omega$. Then for every distinct
  $\alpha, \beta\in\Omega$ there exists $s\in S$ such that
  $\alpha^s=\beta$ and $s$ fixes no point of $\Omega$.

It is an easy exercise using the orbit-counting theorem to show that
if $G$ is a transitive permutation group on a finite set $\Omega$,
then the set of derangements of $G$ generates a transitive subgroup.

It is also easy to see that in general the set of derangements,
together with the identity, need not be transitive on $\Omega$. A
small example is the alternating group $A_4$ acting on the $2$-subsets
of $\{1,2,3,4\}$: An element $g\in A_4$ moving $\{1,2\}$ to
$\{3,4\}$ moves every element in its natural degree $4$ action, so
it is a double transposition, and therefore fixes some $2$-subset.

While there are many more such examples, it seemed hard to find one
under the additional assumption that $G$ acts primitively. In fact the
following Problem 8.75 of the Kourovka Notebook
\cite{kourovka_notebook}, attributed to John G.~Thompson and
designated ``A known problem'', was open for more than 40 years.
\begin{question}\label{q:8.75}
 Suppose $G$ is a finite primitive permutation group on $\Omega$, and
 $\alpha$, $\beta$ are distinct points of $\Omega$. Does there exist
 an element $g\in G$ such that $\alpha^g=\beta$ and $g$ fixes no
 point of $\Omega$?
\end{question}
In \cite{mueller___8.75} we give a negative answer. The group there is
the Steinberg triality group ${}^{3}D_{4}(2)$ of order
$211{,}341{,}312 = 2^{12}\cdot 3^{4}\cdot 7^{2}\cdot 13$ in a
primitive permutation action on a set of size $4{,}064{,}256$.

Using Magma \cite{magma}, we checked that this is the only example if
we assume that $G$ is almost simple with socle of size less than or
equal $20{,}158{,}709{,}760$. It would have been computationally too
expensive to check all the groups with socle $\PSL_2(q)$ in this
range. The purpose of this note is to show that these groups do not
produce negative examples. In fact a somewhat stronger result holds.

In some cases there are lengthy calculations which, in principle,
could be carried out by hand. Instead, we provide SageMath code at
\cite{dera_psl2_verify} that confirms the claims.
\section{Quartic hyperelliptic curves}
If $D$ is a plane algebraic curve over a field $F$ given by
$P(X, Y)\in F[X, Y]$, then $D(F)=\defset{(x, y)\in F^2}{P(x,y)=0}$
denotes the set of affine $F$-points on $D$.

Finally assume that $f(X)=(aX^2-2X+d)(aX^2+2X+d)$ is
 inseparable. The two factors are relatively prime (because the
 difference is $4X$). Furthermore, the discriminant of both factors
 is $4(1-ad)$, hence $ad=1$ and we are done.
\end{proof}
\begin{remark}
 One can check that $q=13$, $17$, and $19$ do not give
 exceptions. However, $q=3$, $5$, $7$, $9$, and $11$ are indeed
 exceptions. This makes it unlikely that there is a ``cleaner'' proof
 of the lemma for odd $q$.
\end{remark}
\begin{corollary}\label{c:split}
 Let $q\ge4$ be a prime power and $a, b, c, d\in\FF_q$ with
 $ad-bc=1$. Set
 $T=\defset{r+1/r}{r\in\Fqs}\cup\set{0}$. Suppose that
 $ar+d/r\in T$ and $-cr+b/r\in T$ for all $r\in\Fqs$. Then one
 of the following holds:
 \begin{itemize}
 \item[(a)] $a=d=0$ or $b=c=0$;
 \item[(b)] $q=9$ and $ar+d/r=2$ for some $r\in\Fqs$.
 \end{itemize}
\end{corollary}
\begin{proof}
 Using the previous lemma, one only need to check the cases where
 $q\le19$ is an odd prime power. This is most conveniently done with
 the SageMath code at \cite{dera_psl2_verify}.
\end{proof}
\begin{lemma}
 Let $q$ be a prime power with $q\ge23$ if $q$ is odd. For
 $F=\FF_{q^2}$ set $N=\defset{r\in F}{r^{q+1}=1}$. Suppose that for
 $0\ne a\in F$ the following holds: For all $r\in N$ with
 $ar+a^q/r\ne0$ there is $s\in N$ such that $ar+a^q/r = s+1/s$. Then
 $a\in N$.
\end{lemma}
\begin{proof} We first handle the easier case that $q$ is even.
As the square map is an automorphism of $F$, there is $r_0\in F$ with
$a^{q-1}=r_0^2$. Note that $a^{q-1}\in N$, hence $r_0\in N$.

We see that the curve $Y^2=\gamma P(X)Q(X)$ has at least $2(q-6)$
 affine points over $\FF_q$. On the other hand, by Lemma
 \ref{l:hasse}, this number is at most $q+1+2\sqrt{q}$. From this one
 quickly obtains $q\le19$.
\end{proof}
\begin{remark}
 One can check that $q=11$, $17$, and $19$ do not give exceptions.
 However, the cases $q=3$, $5$, $7$, $9$, $13$ are indeed
 exceptions. This is most easily seen for $q=3$, because then
 $\defset{s+1/s}{s\in N}=\FF_q$, so the condition on $a$ is vacuous.
\end{remark}
\begin{corollary}\label{c:nonsplit}
 Let $q$ be a prime power. Set $N=\defset{r\in\FF_{q^2}}{r^{q+1}=1}$
 and $T=\defset{r+1/r}{r\in N}$. For $a, b\in\FF_{q^2}$ with
 $a^{q+1} + b^{q+1} = 1$ suppose that $ar+a^q/r\in T$ and
 $br+b^q/r\in T$ for all $r\in\Fqs$. Then $a=0$ or $b=0$.
\end{corollary}
\begin{proof}
 If $q\ge23$, then the previous lemma shows that $a^{q+1}=1$, hence
 $b=0$, or $b^{q+1}=1$, hence $a=0$. The cases $q\le19$ are
 checked in \cite{dera_psl2_verify}.
\end{proof}
\section{The groups of quaternions $2A_4$, $2S_4$, and $2A_5$}
\begin{lemma}
 Let $F$ be a field and $\bH$ be a subgroup of $\PSL_2(F)$ isomorphic
 to $A_4$, $S_4$, or $A_5$. Let $H$ be the preimage of $\bH$ in
 $\SL_2(F)$ and $x, y\in F^{2\times 2}$. If the characteristic of $F$
 is $0$ or relatively prime to $\abs{\bH}$, then
 \[
 \sum_{h\in H}\trace(hy)\cdot\trace(hx)^{2m+1} =\abs{\bH}\cdot c_m\cdot (\trace(x)\cdot\trace(y)-\trace(x\cdot y))\cdot\det(x)^m,
 \]
 where $m\in\set{0, 1}$ if $\bH\cong A_4$, $m\in\set{0, 1, 2}$ if
 $\bH\cong S_4$, $m\in\set{0, 1, 2, 3, 4}$ if $\bH\cong A_5$, and
 $c_0, c_1, c_2, c_3, c_4$ = $1, 2, 5, 14, 42$ are the first five
 Catalan numbers (with index shifted by $1$).
\end{lemma}
\begin{proof}
 We assume that the characteristic of $F$ is $0$ or relatively prime
 to $\abs{\bH}$. Let $\bar F$ be an algebraic closure of $F$. It is
 well known that $\PSL_2(\bar F)$ contains exactly one conjugacy
 class of subgroups isomorphic to $\bH$, see e.g.~\cite[Proposition
 4.1]{beauville___pgl2}. (In fact more is true. By \cite[Proposition
 4.2]{beauville___pgl2} $\PGL_2(F)$ contains at most one conjugacy
 class of subgroups isomorphic to $\bH$.)

Note that for $h\in H$ (recall that $H\le\SL_2(\bar F)$), we
 have $f(h'x, h'y)=f(x, y)$, so $f$ is an invariant of
 $H$. Working out the low degree invariants of $H$ could
 provide a more conceptual explanation. (At a first glance, it
 might not be obvious that the right hand side is an invariant
 under simultaneous left multiplication of $x$ and $y$ by
 $h$. But this follows from the identity
 $\trace(x)\cdot\trace(y)-\trace(x\cdot y)=\trace(\adj(y)\cdot
 x)$, where $\adj(y)$ denotes the adjugate of $y$.)
 \end{itemize}
\end{remark}
\begin{lemma}\label{l:polyhedral}
 Let $F$ be a field and $\bH$ be a subgroup of $\PSL_2(F)$ isomorphic
 to $A_4$, $S_4$, or $A_5$. Assume that the characteristic of $F$ is
 $0$ or relatively prime to $\abs{\bH}$. Let $H$ be the preimage of
 $\bH$ in $\SL_2(F)$. Set
 $T=\defset{\trace(h)}{h\in H}\setminus\set{-2, 2}$, and for
 $x = \begin{pmatrix}a & b\\c & d\end{pmatrix}\in\SL_2(F)$ set
 $y = \begin{pmatrix}0 & 0\\c & d\end{pmatrix}$. Then
 \[
 \sum_{h\in H}\left(\trace(hy)\prod_{t\in T}(\trace(hx)-t)\right) =\abs{\bH}.
 \]
\end{lemma}
\begin{proof}
 With this choice of $x$ and $y$ we have
 \[\trace(x)\cdot\trace(y)-\trace(x\cdot y)=ad-bc=\det(x)=1,\]
 so the previous lemma yields
 \[
 \sum_{h\in H}\trace(hy)\cdot\trace(hx)^{2m+1} =\abs{\bH}\cdot c_m.
 \]
 Let $Z$ be a variable over $F$. Then \cite{dera_psl2_verify} verifies
 \[
 \prod_{t\in T}(Z-t) =
 \begin{cases}
 Z^3 - Z,&\text{ if }\bH\cong A_4\\
 Z^5 - 3Z^3 + 2Z,&\text{ if }\bH\cong S_4\\
 Z^7 - 4Z^5 + 4Z^3 - Z,&\text{ if }\bH\cong A_5.\\
 \end{cases}
 \]
This yields
 \[
 \sum_{h\in H}\left(\trace(hy)\prod_{t\in
 T}(\trace(hx)-t)\right) =
 \begin{cases}
 \abs{\bH}(c_1-c_0)=\abs{\bH},&\text{ if }\bH\cong A_4\\
 \abs{\bH}(c_2-3c_1+2c_0)=\abs{\bH},&\text{ if }\bH\cong S_4\\
 \abs{\bH}(c_3-4c_2+4c_1-c_0)=\abs{\bH},&\text{ if }\bH\cong A_5.\\
 \end{cases}
 \]
\end{proof}
\section{Proof of Theorem \ref{t:main}}
The maximal subgroups of almost simple groups with socle $\PSL_2(q)$
are known; see \cite[Table~8.1, Table~8.2]{bray_etal___maximal} or
\cite{giudici___psl2}. The results in \cite{bray_etal___maximal} and
\cite{giudici___psl2} are more detailed than those in the following
lemma, which lists only the restrictions on $q$ that we shall use.
\begin{lemma}\label{l:giudici}
 Let $q\ge4$ be a prime power and $S=\PSL_2(q)\le
 G\le\PGammaL_2(q)$. Let $M$ be a maximal subgroup of $G$ with
 $S\not\le M$. Set $\bH=M\cap S$. Then one of the following holds.
 \begin{itemize}
 \item[(a)] $[S:\bH] = q+1$,
 \item[(b)] $[S:\bH] = q(q+1)/2$,
 \item[(c)] $[S:\bH] = q(q-1)/2$,
 \item[(d)] $\bH\cong A_4$, and $q$ is and odd prime,
 \item[(e)] $\bH\cong S_4$, and $q$ is a prime with $q\equiv\pm1\mod{8}$,
 \item[(f)] $\bH\cong A_5$, and $q\equiv\pm1\mod{10}$ and $q=9$ if $q$ is
 power of $3$,
 \item[(g)] $\bH\cong\PSL_2(r)$ or $\bH\cong\PGL_2(r)$, where $q=r^e$
 for $e\ge2$.
 \end{itemize}
 Furthermore, in each case $\bH$ is unique up to conjugacy in
 $\aut(S)$ (with two classes in case (g) if $r$ is odd).
\end{lemma}
Let $S$ and $G$ be as in Theorem \ref{t:main}, $M$ be a stabilizer of
a point in $\Omega$, and $\bH=M\cap S$. Then $M$ is a maximal subgroup
of $G$ by primitivity. We handle the cases (a) to (g) separately.

The remaining cases (b) to (g) require some work. Suppose that there
is a counterexample. Then there exist distinct
$\alpha, \beta\in\Omega$ such that no derangement in $S$ moves
$\alpha$ to $\beta$. Let $\bH$ be the stabilizer in $S$ of $\alpha$
and $\bx\in S$ with $\alpha^{\bx}=\beta$. Then $\bH\bx$ is the set of
elements which moves $\alpha$ to $\beta$. As this set does not contain
a derangement, every element in the coset $\bH\bx$ is conjugate to an
element from $\bH$.

If $G$ acts primitively on $\Omega$, then the possibilities for a point
stabilizer $S_\omega < S$ are known. While some cases are relatively
easy to handle, the more difficult ones arise when $S_\omega$ is
dihedral, or when $S_\omega$ is one of the polyhedral groups $A_4$,
$S_4$, or $A_5$. In the former case we require Hasse’s bound on the
number of points on elliptic curves over finite fields, and to rule out
the latter cases we derive trace identities for the binary polyhedral
groups $2A_4$, $2S_4$, and $2A_5$.

In some cases there are lengthy calculations which, in principle,
could be carried out by hand. Instead, we provide SageMath code at
\cite{dera_psl2_verify} that confirms the claims.

To prove Theorem~\ref{t:main}, one uses that when $G$ acts primitively on $\Omega$, “the possibilities for a point stabilizer $S_\omega < S$ are known.” The argument then splits by the type of $S_\omega$: “some cases are relatively easy to handle,” while the “more difficult ones arise when $S_\omega$ is dihedral, or when $S_\omega$ is one of the polyhedral groups $A_4$, $S_4$, or $A_5$.” In the dihedral case, the proof “require[s] Hasse’s bound on the number of points on elliptic curves over finite fields,” and to exclude the polyhedral cases one “derive[s] trace identities for the binary polyhedral groups $2A_4$, $2S_4$, and $2A_5$.” The authors note that “in some cases there are lengthy calculations” and they “provide SageMath code at \cite{dera_psl2_verify} that confirms the claims.”

To prove the main theorem, one uses that when $G$ acts primitively on $\Omega$, “the possibilities for a point stabilizer $S_\omega < S$ are known.” The argument then splits by the type of $S_\omega$: “some cases are relatively easy to handle,” while the “more difficult ones arise when $S_\omega$ is dihedral, or when $S_\omega$ is one of the polyhedral groups $A_4$, $S_4$, or $A_5$.” In the dihedral case, the proof “require[s] Hasse’s bound on the number of points on elliptic curves over finite fields,” and to exclude the polyhedral cases one “derive[s] trace identities for the binary polyhedral groups $2A_4$, $2S_4$, and $2A_5$.” The authors note that “in some cases there are lengthy calculations” and they “provide SageMath code at \cite{dera_psl2_verify} that confirms the claims.”

Question: Let $q\ge 4$ be a prime power, let $S=\mathrm{PSL}_2(q)$, and let $G$ be an almost simple group with socle $S$, meaning that $S\le G\le \mathrm{Aut}(S)$. Suppose $G$ acts primitively on a set $\Omega$. For distinct points $\alpha,\beta\in\Omega$, which statement holds about elements of $S$ in this permutation action?

For every pair of distinct points $\alpha,\beta\in\Omega$, there exists an element $s\in S$ such that $\alpha^s=\beta$ and $s$ fixes no point of $\Omega$; equivalently, $S$ contains a derangement sending $\alpha$ to $\beta$.

Choices:
B. For every pair of distinct points $\alpha,\beta\in\Omega$, there exists an element $s\in G$ such that $\alpha^s=\beta$ and $s$ fixes no point of $\Omega$; equivalently, $G$ contains a derangement sending $\alpha$ to $\beta$.
C. For every pair of distinct points $\alpha,\beta\in\Omega$, there exists an element $s\in S$ such that $\alpha^s=\beta$.
D. For every point $\alpha\in\Omega$, there exists an element $s\in S$ such that $s$ fixes no point of $\Omega$ and $\alpha^s\ne\alpha$; equivalently, for each $\alpha$ there is some derangement in $S$ moving $\alpha$.
E. There exists a single element $s\in S$ such that $s$ fixes no point of $\Omega$ and, for every ordered pair of distinct points $\alpha,\beta\in\Omega$, one has $\alpha^s=\beta$.


\begin{rem}\label{rem:DMexs}
Some nonarithmetic Deligne--Mostow lattices \cite{DeligneMostow} also admit homomorphisms onto cocompact Fuchsian groups from \emph{forgetful maps} associated with contracting weights. See for example \cite[Thm.\ 3.1(iv)]{Deraux}.
\end{rem}

\subsection{Automorphisms of Fuchsian groups}\label{ssec:FuchsianAut}

The purpose of this subsection is to prove \Cref{thm:Fixpsubs}.

\begin{pf}[Proof of \Cref{thm:Fixpsubs}]
Let $p$ be a prime, $T$ be a surface group of genus $g \ge2$, and $\phi \in \Aut(T)$ have the property that $\phi(N) = N$ for every finite index normal subgroup with index a power of $p$. Consider the pro-$p$ completion $T_{\wh{p}}$ of $T$. Then $\varphi$ extends to a continuous automorphism of $T_{\wh{p}}$ that is \emph{normal}, i.e., so that $\varphi(\wh{N})$ for every closed normal subgroup of $T_{\wh{p}}$. Indeed, every such normal subgroup is an intersection of finite index normal subgroups of $T_{\wh{p}}$, and finite index normal subgroups of $T_{\wh{p}}$ are in natural one-to-one correspondence with the normal $p$-power index subgroups of $T$.

Then $T_{\wh{p}}$ is topologically finitely presented, since $T$ is finitely presented, hence $T_{\wh{p}}$ is \emph{psuedo $p$-free} in the sense of Jarden--Ritter (see \cite[\S 1]{JardenRitter} for the definition) by \cite[Thm. 4]{JardenRitter}. Thus the extension of $\varphi$ to $T_{\wh{p}}$ is inner by \cite[Thm.\ 1]{JardenRitter}. Moreover, Paris proved that $T$ is $p$-conjugacy separable \cite[Thm.\ 1.7]{Paris}, which implies that $\varphi$ must preserve conjugacy classes of $T$ (i.e., it is \emph{pointwise inner}). Therefore $\varphi$ is a normal automorphism of $T$, meaning that it preserves all normal subgroups of $T$. Finally, it is a theorem of Bogopolski, Kudryavtseva, and Zieschang that normal automorphisms of higher-genus surface groups are inner \cite[Thm.\ 1.3]{BKZ}, which completes the proof of the theorem.
\end{pf}

\section{The main result}\label{sec:Main}

Suppose that $\Lam < \PU(2,1)$ is a cocompact nonarithmetic lattice with nontrivial universal homomorphism $\rho$. Following \Cref{ssec:PU21}, let $\Gam < \PU(2,1)$ be the unique maximal lattice containing $\Lam$. By \Cref{lem:TFFuchsian} and by passing to a finite index subgroup if necessary, we can assume that the universal homomorphism for $\Lam$ has a torsion-free factor and that $\Lam$ is a characteristic subgroup of $\Gam$. If $M_{\mathrm{tf}} \coloneqq \ker(\rho_{\mathrm{tf}})$ denotes the kernel of the torsion-free universal homomorphism for $\Lam$, then $M_{\mathrm{tf}}$ is an infinite index characteristic subgroup of $\Gam$ by \Cref{lem:UniversalChar} and because a characteristic subgroup of a characteristic subgroup is characteristic in the larger group. Set $T \coloneqq \Lam / M_{\mathrm{tf}}$, so $T = \rho_{\mathrm{tf}}(\Lam)$.

As described in \Cref{ssec:PU21}, away from finitely many prime ideals $\frakp$ of the ring $\Tr\Ad \Gam$, there is a well-defined congruence quotient $\wh{Q} \coloneqq \Gam / \Gam(\frakp)$ with trivial center and commutator subgroup $Q$ that is a nonabelian finite simple group. Moreover, $\Lam$ and $M_{\mathrm{tf}}$ are Zariski dense in $\PU(2,1)$, hence after possibly passing to a finite index subgroup of $\Lam$, we can assume that they both map onto $Q$ for infinitely many $\frakp$. Fix one such $\frakp$ for the remainder of this section. Then define $\Lam(\frakp) \coloneqq \Gam(\frakp) \cap \Lam$ and $M_{\mathrm{tf}}(\frakp) \coloneqq \Gam(\frakp) \cap M_{\mathrm{tf}}$, noting that both $\Lam(\frakp)$ and $M_{\mathrm{tf}}(\frakp)$ are normal subgroups of $\Gam$, since each is the intersection of normal subgroups of $\Gam$.

\begin{lem}\label{lem:QuoSetup}
With the notation established in this section,
\begin{align*}
\rho_{\mathrm{tf}}(\Lam(\frakp)) &= T \\
\Lam / M_{\mathrm{tf}}(\frakp) &\cong Q \times T \\
\Gam / M_{\mathrm{tf}}(\frakp) &\cong \wh{Q} \times \wh{T}
\end{align*}
where $\wh{T} \coloneqq \Gam / M_{\mathrm{tf}}$ contains $T$ as a finite index subgroup.
\end{lem}
\begin{pf}
The first equality holds since $M_{\mathrm{tf}}$ contains coset representatives for $\Lam(\frakp)$ in $\Lam$. The second holds since $M_{\mathrm{tf}} / M_{\mathrm{tf}}(\frakp)$ and $\Lam(\frakp) / M_{\mathrm{tf}}(\frakp)$ are disjoint normal subgroups of $\Lam / M_{\mathrm{tf}}(\frakp)$, and the third is similar.
\end{pf}

\begin{rem}
Note that $T$ is then a product of factors of the torsion-free universal homomorphism of $\Lam(\frakp)$, but there may be additional factors.
\end{rem}

\begin{equation}\label{eq:ReductionModp}
\pi_\frakp : \wt{\bfG}(\calO_k) \lra \wt{\bfG}(\bbF_\frakp)
\end{equation}
to any $\wt{\Del} \le \wt{\bfG}(\calO_k)$ has kernel the \emph{congruence subgroup} $\wt{\Del}(\frakp) \trianglelefteq \wt{\Del}$. Strong approximation for Zariski dense subgroups of simply connected algebraic groups says that if $\wt{\Del}$ is Zariski dense in $\wt{\bfG}(k)$, then $\pi_\frakp(\wt{\Del}) = \wt{\bfG}(\bbF_\frakp)$ for all but finitely many $\frakp$ \cite[Thm.\ 1.1]{Weisfeiler}; also see Cor.\ 3 in Window 9 of \cite{LubotzkySegal} along with discussion on p.\ 392 implying the more general statement for number fields other than $\bbQ$. If $\wt{\Del}$ is the preimage of $\Del$ in $\SU(2,1)$, then congruence subgroups of $\Del$ are those whose preimage in $\wt{\Del}$ are congruence subgroups.

Since $\PU(2,1)$ has type ${}^2 \mathrm{A}_2$, there is a quadratic extension $\ell$ of $k$ canonically associated with $\bfG$. If $\Del \le \Gam$ is Zariski dense in $\PU(2,1)$, then away from a finite set of exceptional primes, the group $\Del / \Del(\frakp)$ has commutator subgroup $\PSL_3(\bbF_\frakp)$ if $\frakp$ splits in $\ell$ or $\PU(3; \bbF_\frakp)$ if $\frakp$ is inert in $\ell$. In particular, the congruence quotient of $\Del$ has commutator subgroup a nonabelian finite simple group for all but finitely many primes $\frakp$. For example in the split case, observe that $\Del / \Del(\frakp)$ is then naturally a subgroup of $\PGL_3(\bbF_\frakp)$ containing $\PSL_3(\bbF_\frakp)$, so it has trivial center.

\subsection{K\"ahler groups}\label{ssec:KahlerGroups}

A \emph{K\"ahler group} is the fundamental group of a compact K\"ahler manifold. See \cite{KahlerBook, Py} for basics on K\"ahler groups. Note that finite index subgroups of K\"ahler groups are naturally K\"ahler groups.

Let $\Gam < \PU(2,1)$ be a cocompact lattice, possibly with torsion. Since $\PU(2,1)$ is a linear Lie group and $\Gam$ is finitely generated, $\Gam$ is residually finite and thus there is a finite index subgroup $\Lam < \Gam$ that is torsion-free. Since smooth projective varieties are compact K\"ahler manifolds, by pullback of the Fubini--Study metric on $\bbP^N$ under a projective embedding, it follows that any such $\Lam$ is a K\"ahler group. It then follows from \cite[Lem.\ 1.1.15]{KahlerBook} that all cocompact lattices in $\PU(2,1)$ are K\"ahler groups.

\medskip

Fundamental to this paper is the \emph{universal homomorphism} attached to any K\"ahler group. See \cite[\S 2.C]{HLPSV} for discussion and references for the following result defining the universal homomorphism.

\begin{thm}\label{thm:Universal}
Suppose $\Lam$ is a K\"ahler group. Then there is a (possibly empty) finite collection of cocompact Fuchsian groups $\{T_\al\}_{\al \in \frakU}$ and a homomorphism
\begin{equation}\label{eq:UniversalHom}
\rho : \Lam \lra T \coloneqq \prod_{\al \in \frakU} T_\al
\end{equation}

\label{sec:Intro}

Throughout this paper, an automorphism of a complex manifold will always mean a biholomorphism. The main result of this paper, proved in \Cref{sec:Main}, is the following.

\begin{abstract}
In response to a question raised by Belolipetsky and the first author in \cite{BelolipetskyLubotzky}, we prove that for every finite group $G$ there are infinitely many isomorphism classes of compact complex hyperbolic $2$-manifolds with automorphism group isomorphic to $G$.
\end{abstract}

Throughout this paper, an automorphism of a complex manifold will always mean a biholomorphism. The main result of this paper, proved in \Cref{sec:Main}, is the following.

Greenberg \cite{Greenberg} proved \Cref{thm:Main} for Riemann surfaces, i.e., complex hyperbolic $1$-manifolds. Belolipetsky and the first author \cite{BelolipetskyLubotzky} proved the analogue of \Cref{thm:Main} for isometry groups of hyperbolic $n$-manifolds, $n \ge 3$, and this paper answers a question raised in \cite[\S 6.5]{BelolipetskyLubotzky}. Our argument follows the broad outline of that paper, but differences at key steps create many new challenges we must overcome.

\begin{rem}\label{rem:Highern}
Whether or not there are cocompact nonarithmetic lattices in $\PU(n,1)$ for all $n \ge 3$ is one of the major open problems in the study of lattices in Lie groups. For any $n \ge 3$, if there is a cocompact nonarithmetic lattice in $\PU(n,1)$ with nontrivial universal homomorphism, then this paper produces infinitely many isomorphism classes of closed complex hyperbolic $n$-manifolds with isometry group any finite group $G$ \emph{verbatim}.
\end{rem}

\begin{thm}\label{thm:ProjectiveVersion}
For every finite group $G$, there are infinitely many isomorphism classes of irreducible, minimal smooth complex projective surfaces of general type with automorphism group $G$.
\end{thm}

In what follows, $\pi : T \to T_r$ will denote projection onto the $r^{th}$ surface group factor of $T$ for an arbitrary $r$ that will be fixed for the remainder of this section. Let $\sig$ denote projection of $\Gam$ onto $\Gam / M_{\mathrm{tf}}(\frakp)$ and also the restriction of $\sig$ to any finite index subgroup of $\Gam$. This leads to the diagram of groups and surjective homomorphisms given in \Cref{fig:BigDiagram}.
\begin{figure}[h]
\centering
\begin{tikzcd}
& \Gam \arrow[ddr, -] \arrow[dl, -] \arrow[rr, ->, "\sig"] \arrow[rrrr, bend left = 35, ->, "\textrm{mod}\, M_{\mathrm{tf}}"] & & \wh{Q} \times \wh{T} \arrow[d, -] \arrow[rr, ->, "\textrm{mod}\, \wh{Q}" below] & & \wh{T} \arrow[d, -] \\
\Gam(\frakp) \arrow[dd, -] & & & Q \times T \arrow[rr, ->, "\textrm{mod}\, Q"] && T \arrow[rr, ->, "\pi"] & & T_r \\
& & \Lam \arrow[dll, -] \arrow[ur, ->, "\sig"] \arrow[urrr, bend right=20, ->, "\rho_{\mathrm{tf}}" {xshift=0.75em, yshift=-1.25em}] \\
\Lam(\frakp) \arrow[ddd, -] \arrow[uurrrrr, ->, bend right=35, "\sig\, =\, \rho_{\mathrm{tf}}|_{\Lam(\frakp)}" {xshift=6em, yshift=-0.5em}] & & \\
& & \\
& & M_{\mathrm{tf}} \arrow[dll, -] \arrow[uuu, -, crossing over] & \\
M_{\mathrm{tf}}(\frakp) & &
\end{tikzcd}
\caption{The relationships between various groups}\label{fig:BigDiagram}
\end{figure}
Define
\begin{equation}\label{eq:TauDef}
\tau \coloneqq \pi \circ \rho_{\mathrm{tf}} : \Lam \lra T_r
\end{equation}
and $M \coloneqq \ker(\tau|_\Del)$. For a finite index subgroup $\Del \le \Gam$, let $N_\Gam(\Del)$ denote its normalizer in $\Gam$. If $\Del$ contains $M$, then any $\gam \in N_\Gam(\Del)$ normalizing $M$ induces an automorphism of $\Del / M$ through $\pi \circ \rho_{\mathrm{tf}}$. The following proposition/construction is essential for proving \Cref{thm:Main}.

\begin{pf}[Proof of \Cref{thm:Main}]
The only input to begin this section of the paper was a cocompact nonarithmetic lattice $\Lam < \PU(2,1)$ with nontrivial universal homomorphism. Such examples exist by \Cref{thm:Examples}. If $G$ is a nontrivial group (resp.\ the trivial group), then \Cref{prop:FinalNontrivialG} (resp.\ \Cref{prop:TrivialCase}) produces a torsion-free lattice $B \le \Lam$ whose normalizer $N(B) < \PU(2,1)$, which equals $N_\Gam(B)$ for $\Gam$ the unique maximal lattice in $\PU(2,1)$ containing $B$ by the Margulis dichotomy, has $N(B) / B \cong G$. Mostow rigidity now implies that $G$ is the automorphism group (equivalently, holomorphic isometry group) of the smooth compact complex hyperbolic $2$-manifold $B \bs \bbB^n$. Varying the primes $\{p_j\}$ in our construction gives infinitely many lattices with covolume tending to infinity for any fixed finite group $G$, hence another application of Mostow rigidity implies that there are infinitely many isomorphism classes of lattices. This proves the theorem.
\end{pf}

Complex hyperbolic manifolds are of general type \cite[\S V.20]{BPV}, hence \Cref{thm:Main} immediately implies \Cref{thm:ProjectiveVersion}. Surfaces of general type all have finite automorphism group, which is directly analogous to the classical Hurwitz bound $|\Aut(C)| \le 84(g-1)$ for a smooth projective curve of genus $g$. In fact, $|\Aut(X)|$ can be bounded from above in terms of the self-intersection of the first Chern class of the canonical bundle; see for example \cite{Xiao}. Thus it is natural that the volume (or Euler characteristic, by Gauss--Bonnet) goes to infinity with $|G|$ for our examples.

\begin{thm}\label{thm:Examples}
There are cocompact nonarithmetic lattices $\Gam$ in $\PU(2,1)$ with nontrivial universal homomorphism.
\end{thm}

\begin{thm}\label{thm:Main}
For every finite group $G$, there are infinitely many isomorphism classes of closed complex hyperbolic $2$-manifolds with automorphism group isomorphic to $G$.
\end{thm}

Greenberg \cite{Greenberg} proved \Cref{thm:Main} for Riemann surfaces, i.e., complex hyperbolic $1$-manifolds. Belolipetsky and the first author \cite{BelolipetskyLubotzky} proved the analogue of \Cref{thm:Main} for isometry groups of hyperbolic $n$-manifolds, $n \ge 3$, and this paper answers a question raised in \cite[\S 6.5]{BelolipetskyLubotzky}. Our argument follows the broad outline of that paper, but differences at key steps create many new challenges we must overcome.

Recall that a compact complex hyperbolic $n$-manifold is a quotient $\Gamma \backslash \mathbb{B}^n$ of the unit ball in $\mathbb{C}^n$ by a torsion-free cocompact lattice $\Gamma$ in the Lie group $\PU(n,1)$. Note that $\mathbb{B}^1$ is simply the Poincar\'e disk with the hyperbolic metric. For $n \ge 2$, Mostow rigidity then implies that the automorphism group of a complex hyperbolic manifold is precisely the holomorphic isometry group, and the proof of \Cref{thm:Main} is through this interpretation.

One strong similarity between the proofs is in the input: a cocompact nonarithmetic lattice in the isometry group of the symmetric space. Moreover, \cite{BelolipetskyLubotzky} uses a nonabelian free quotient to begin the construction, whereas we use a quotient that is a cocompact Fuchsian group. The subgroup counting methods in both papers are nearly identical. Note that it is quite easy, given a free or surface group quotient of a finite index subgroup $\Lambda$ of a lattice $\Gamma$, to produce a finite index subgroup $B$ of $\Gamma$ so that $N_\Gamma(B) / B$ \emph{contains} $G$, where $N_\Gamma(B)$ is the normalizer of $B$ in $\Gamma$, but ensuring that $B$ is torsion-free and that $N_\Gamma(B) / B$ is \emph{exactly} $G$ is quite subtle.

The nonarithmetic lattices used in \cite{BelolipetskyLubotzky} are the famous examples of Gromov and Piatetski-Shapiro \cite{GPS}. These lattices naturally have the structure of a free product with amalgamation over the fundamental group of a codimension one totally geodesic subspace, and the nonabelian free quotient of a finite index subgroup is derived from this (see \cite{LubotzkyFreeQuo}). In the complex hyperbolic setting, there are no real codimension one totally geodesic subspaces \cite[\S 3.1]{Goldman}, hence no such splittings.

Our replacement, from the theory of K\"ahler groups, is the \emph{universal homomorphism} to a direct product of cocompact Fuchsian groups; see \cite{HLPSV} for an introduction. Cocompact nonarithmetic lattices in $\PU(2,1)$ with nontrivial universal homomorphism are found among the famous examples of Livn\'e \cite{Livne} and Deligne--Mostow \cite{DeligneMostow}; see \Cref{thm:Examples}. With this different input, a key lattice $\Delta$ constructed early in the outline of \cite{BelolipetskyLubotzky} now fails to be normal in the maximal lattice $\Gamma$ in $\PU(2,1)$. In a certain sense, much of the work in \Cref{sec:Main} is using properties of the universal homomorphism to circumvent this significant obstruction to executing several steps in the strategy from \cite{BelolipetskyLubotzky}.

\begin{rem}\label{rem:Highern}
Whether or not there are cocompact nonarithmetic lattices in $\PU(n,1)$ for all $n \ge 3$ is one of the major open problems in the study of lattices in Lie groups. For any $n \ge 3$, if there is a cocompact nonarithmetic lattice in $\PU(n,1)$ with nontrivial universal homomorphism, then this paper produces infinitely many isomorphism classes of closed complex hyperbolic $n$-manifolds with isometry group any finite group $G$ \emph{verbatim}.
\end{rem}

In this paper, a genus $g$ surface group will always mean the fundamental group of a closed Riemann surface of genus $g$, despite the slight clash of terminology with fundamental groups of complex surfaces also appearing. In \Cref{ssec:FuchsianAut} we will prove the following result on automorphisms of surface groups, which is an analogue of a theorem of the first author in the free group case \cite[Thm.\ 1]{LubotzkyFree} that may be of independent interest.

\begin{thm}\label{thm:Fixpsubs}
Let $T$ be a surface group of genus $g \ge 2$, $p$ be a prime, and suppose that $\varphi \in \Aut(T)$ has the property that $\varphi(N) = N$ for all finite index normal subgroups of $T$ with index a power of $p$. Then $\varphi$ is inner.
\end{thm}

\noindent\textbf{Interpretation as a complex projective surface}

\medskip

Compact complex hyperbolic $2$-manifolds are also minimal smooth complex projective surfaces of general type \cite[\S V.20]{BPV}. The appendix to this paper recasts \Cref{thm:Main} in that context to prove the following result.

\begin{thm}\label{thm:ProjectiveVersion}
For every finite group $G$, there are infinitely many isomorphism classes of irreducible, minimal smooth complex projective surfaces of general type with automorphism group $G$.
\end{thm}

A surface is irreducible if it is not finitely (\'etale) covered by a product of complex curves. Greenberg's theorem easily produces products of complex curves with any finite automorphism group, so only the irreducible case is of interest. As described in the appendix, one may be able to construct irreducible examples using other methods. For example, there may exist a nonsingular fibration $X \to C$ over a curve $C$ (e.g., a ruled surface) where $\Aut(C) \cong G$ by Greenberg and there is a natural isomorphism $\Aut(X) \cong \Aut(C)$. However, we were not able to find a reference for \Cref{thm:ProjectiveVersion} in the literature.

\begin{rem}\label{rem:HardPart}
Our construction is also related to a holomorphic map to a smooth projective curve (see \Cref{ssec:KahlerGroups}). However, the bulk of the proof is precisely in showing that our surface has no extra automorphisms, which is highly unclear from the nature of the construction.
\end{rem}

\medskip

\noindent\textbf{Organization of the paper}

\medskip

\noindent
This paper is organized as follows. \Cref{sec:Prelim} contains preliminaries about lattices in $\PU(2,1)$, universal homomorphisms for K\"ahler groups, and the proof of \Cref{thm:Fixpsubs}. \Cref{sec:Main} proves \Cref{thm:Main}. The appendix discussions automorphism groups of smooth complex projective surfaces.

\medskip

\noindent \textbf{Acknowledgments}

\medskip

\noindent
Much of the work on this paper was done while the authors were at the conference `Discrete and profinite groups' at the Isaac Newton Institute for Mathematical Sciences, and thank the institute for its hospitality. The first author was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 (N.\ 882751). The second author was supported by Grants DMS-2203555 and DMS-2506896 from the National Science Foundation and the Simons Foundation [SFI-MPS-TSM-00014184, MS].

The proof of \Cref{thm:Main} is “through” the fact that for $n\ge 2$, Mostow rigidity implies “the automorphism group of a complex hyperbolic manifold is precisely the holomorphic isometry group.” The argument “follows the broad outline” of \cite{BelolipetskyLubotzky}: start from “a cocompact nonarithmetic lattice” in the relevant isometry group, use a quotient of a finite index subgroup $\Lambda<\Gamma$ (here “a quotient that is a cocompact Fuchsian group,” rather than a free quotient), and then apply “subgroup counting methods” (said to be “nearly identical” to \cite{BelolipetskyLubotzky}) to build finite-index subgroups $B<\Gamma$.

A key step is to arrange, for such $B$, that the normalizer quotient realizes the desired finite group: it is “quite easy…to produce a finite index subgroup $B$ of $\Gamma$ so that $N_\Gamma(B) / B$ \emph{contains} $G$,” but “ensuring that $B$ is torsion-free and that $N_\Gamma(B) / B$ is \emph{exactly} $G$ is quite subtle,” and the “bulk of the proof is precisely in showing that our surface has no extra automorphisms.”

Because in the complex hyperbolic setting “there are no real codimension one totally geodesic subspaces…hence no such splittings,” the paper replaces the free-quotient input by the “universal homomorphism to a direct product of cocompact Fuchsian groups.” With this different input, “a key lattice $\Delta$…now fails to be normal in the maximal lattice $\Gamma$,” and “much of the work in \Cref{sec:Main} is using properties of the universal homomorphism to circumvent this significant obstruction to executing several steps in the strategy from \cite{BelolipetskyLubotzky}.”

To prove the main theorem, the argument is “through” the fact that for $n\ge 2$, Mostow rigidity implies “the automorphism group of a complex hyperbolic manifold is precisely the holomorphic isometry group.” The argument “follows the broad outline” of \cite{BelolipetskyLubotzky}: start from “a cocompact nonarithmetic lattice” in the relevant isometry group, use a quotient of a finite index subgroup $\Lambda<\Gamma$ (here “a quotient that is a cocompact Fuchsian group,” rather than a free quotient), and then apply “subgroup counting methods” (said to be “nearly identical” to \cite{BelolipetskyLubotzky}) to build finite-index subgroups $B<\Gamma$.

A key step is to arrange, for such $B$, that the normalizer quotient realizes the desired finite group: it is “quite easy…to produce a finite index subgroup $B$ of $\Gamma$ so that $N_\Gamma(B) / B$ \emph{contains} $G$,” but “ensuring that $B$ is torsion-free and that $N_\Gamma(B) / B$ is \emph{exactly} $G$ is quite subtle,” and the “bulk of the proof is precisely in showing that our surface has no extra automorphisms.”

Because in the complex hyperbolic setting “there are no real codimension one totally geodesic subspaces…hence no such splittings,” the paper replaces the free-quotient input by the “universal homomorphism to a direct product of cocompact Fuchsian groups.” With this different input, “a key lattice $\Delta$…now fails to be normal in the maximal lattice $\Gamma$,” and “much of the work later is using properties of the universal homomorphism to circumvent this significant obstruction to executing several steps in the strategy from \cite{BelolipetskyLubotzky}.”

Question: Let an automorphism of a complex manifold mean a biholomorphism. For an arbitrary finite group $G$, which statement holds about closed complex hyperbolic $2$-manifolds (that is, compact complex manifolds of complex dimension $2$ carrying a complex hyperbolic structure)?

There exist infinitely many isomorphism classes of closed complex hyperbolic $2$-manifolds whose automorphism group is isomorphic to $G$.

Choices:
B. There exist infinitely many isomorphism classes of closed complex hyperbolic $2$-manifolds whose automorphism group contains a subgroup isomorphic to $G$.
C. There exists at least one closed complex hyperbolic $2$-manifold whose automorphism group is isomorphic to $G$.
D. There exists a closed complex hyperbolic $2$-manifold $X$ such that every finite group $G$ occurs as a subgroup of $\operatorname{Aut}(X)$.
E. For every finite group $G$, there are infinitely many isomorphism classes of closed complex hyperbolic $2$-manifolds whose full holomorphic isometry group is isomorphic to $G$.


\begin{teorema}\label{thm:main}
Let $\mu$ be a signed Radon measure on $\mathbb{R}^n$ and $k \in \{0,1,\dots,n\}$. Then $\mathcal{S}_\mu^k$ is contained in a countable union of $k$-dimensional Lipschitz graphs. Moreover, the approximate tangent plane to $\mathcal{S}_\mu^k$ at $x$ coincides with $D_x$ for $\mathcal{H}^{k}$-a.e.\ $x \in \mathcal{S}_\mu^k$.
\end{teorema}

The notion of blow-up of a geometric or analytic object, i.e.\ the limit of its enlargements around a point as the zooming factor goes to infinity, is of fundamental importance in many areas of Mathematics. The blow-up of a function or a manifold provides a simplified, but somewhat essential, description of the local behavior of the object ``at infinitesimal scale".
For this and other reasons, the classification of the blow-ups of various mathematical items has become a central theme in geometric and functional analysis.

For a Radon measure $\mu$ in $\mathbb{R}^n$, a natural definition of blow-up at a point $x \in \mathbb{R}^n$ is the one in the sense of Preiss, \cite{preiss87}: the limit in the weak*-topology of Radon measures (in duality with $C_c(\mathbb{R}^n)$) of the rescalings
\begin{equation}\label{eq:rescalings}
c_j (\tau_{x,r_j})_\# \mu,
\end{equation}
for some sequences $c_j>0$, $r_j \to 0$, where $\tau_{x,r}(y)\coloneqq \frac{y-x}{r}$ and ${}_\#$ is the push-forward of Radon measures.

We say that $\mu$ has unique blow-up at $x \in \operatorname{supp} \mu$ if there exists a Radon measure $\mu_x$ with either $|\mu_x|(B_1(0)) = 1$ or $\mu_x = 0$ and such that, for every sequence $r_j \to 0^+$, there are a subsequence, not relabeled, and a constant $c\neq 0$ for which
\begin{equation}\label{eq:def-unique_blowup}
\mu_{x,r_j} \coloneqq
\frac{1}{|\mu|(B_{r_j}(x))} (\tau_{x,r_j})_\# \mu
\overset{\ast}{\rightharpoonup} c \mu_x,
\end{equation}
where $|\mu|$ is the total variation measure of $\mu$ and $\overset{\ast}{\rightharpoonup}$ denotes the weak*-convergence of Radon measures.
We will denote by $\mathcal{S}_\mu$ the set of points where $\mu$ has unique blow-up.

Mattila studied positive measures having a unique blow-up at almost every point: a way to rephrase his main result \cite[Theorem 3.2]{mattila_unique05} is that,
for $\mu$-a.e.\ $x \in \mathcal{S}_\mu$, the blow-up of $\mu$ at $x$ is a multiple of the $k$-dimensional Hausdorff measure $\mathcal{H}^k$ on a $k$-plane.
Although this describes the blow-ups of $\mu$ at $\mu$-a.e.\ point in $\mathcal{S}_\mu$, it can be interesting to study the behavior of $\mu$ in the remaining part of $\mathcal{S}_\mu$.

This problem has already received some attention: in \cite{delnin21} it is proved that, for a given $u \in L^1_{\text{loc}}(\mathbb{R}^n)$, the set $\Sigma_u$ of points $x$ where $\frac{1}{r^n}u \left (\frac{\cdot-x}{r} \right )$ converges, in the $L^1_{\mathrm{loc}}$-topology, to a non-constant function can be covered by a countable union of $(n-1)$-dimensional Lipschitz graphs.
\begin{itemize}
\item The results in \cite{delnin21} rely on the fact that the measure $\mu$ is induced by a $L^1_{\mathrm{loc}}$ function $u$ and that the rescalings converge in the $L^1_{\mathrm{loc}}$-topology, while in $\mathcal{S}_\mu$ the convergence is intended in the weak*-topology of Radon measures. Thus the subset of $\mathcal{S}_\mu$ where the unique blow-up of $u$ is not a constant function can be larger than $\Sigma_u$.
Is this set rectifiable as well? What can be said if $\mu$ is a general Radon measure?

\item It is easily seen (Lemma \ref{lmm:0-hom}) that the unique blow-up $\mu_x$ of $\mu$ at a point in $x \in \mathcal{S}_\mu$ has an \emph{invariant linear subspace} $D_x$ of $\mathbb{R}^n$, namely $(\tau_{y,1})_\# \mu_x = \mu_x$ for $y \in D_x$. Therefore
\begin{equation}\label{eq:def_s_mu_k}
\mathcal{S}_\mu = \bigcup_{k=1}^n \mathcal{S}_\mu^k,
\qquad\text{where} \qquad
\mathcal{S}_\mu^k \coloneqq \left \{x \in \mathcal{S}_\mu \colon \dim D_x=k \right \}.
\end{equation}
Do these $\mathcal{S}_\mu^k$ enjoy finer rectifiability properties?
\end{itemize}

The main result of this note provides an answer to these questions.

For a Radon measure $\mu$ in $\mathbb{R}^n$, a natural definition of blow-up at a point $x \in \mathbb{R}^n$ is the one in the sense of Preiss, \cite{preiss87}: the limit in the weak*-topology of Radon measures (in duality with $C_c(\mathbb{R}^n)$) of the rescalings
\begin{equation}\label{eq:rescalings}
c_j (\tau_{x,r_j})_\# \mu,
\end{equation}
for some sequences $c_j>0$, $r_j \to 0$, where $\tau_{x,r}(y)\coloneqq \frac{y-x}{r}$ and ${}_\#$ is the push-forward of Radon measures.

We say that $\mu$ has unique blow-up at $x \in \operatorname{supp} \mu$ if there exists a Radon measure $\mu_x$ with either $|\mu_x|(B_1(0)) = 1$ or $\mu_x = 0$ and such that, for every sequence $r_j \to 0^+$, there are a subsequence, not relabeled, and a constant $c\neq 0$ for which
\begin{equation}\label{eq:def-unique_blowup}
\mu_{x,r_j} \coloneqq
\frac{1}{|\mu|(B_{r_j}(x))} (\tau_{x,r_j})_\# \mu
\overset{\ast}{\rightharpoonup} c \mu_x,
\end{equation}
where $|\mu|$ is the total variation measure of $\mu$ and $\overset{\ast}{\rightharpoonup}$ denotes the weak*-convergence of Radon measures.
We will denote by $\mathcal{S}_\mu$ the set of points where $\mu$ has unique blow-up.

Mattila studied positive measures having a unique blow-up at almost every point: a way to rephrase his main result \cite[Theorem 3.2]{mattila_unique05} is that,
for $\mu$-a.e.\ $x \in \mathcal{S}_\mu$, the blow-up of $\mu$ at $x$ is a multiple of the $k$-dimensional Hausdorff measure $\mathcal{H}^k$ on a $k$-plane.
Although this describes the blow-ups of $\mu$ at $\mu$-a.e.\ point in $\mathcal{S}_\mu$, it can be interesting to study the behavior of $\mu$ in the remaining part of $\mathcal{S}_\mu$.

This problem has already received some attention: in \cite{delnin21} it is proved that, for a given $u \in L^1_{\text{loc}}(\mathbb{R}^n)$, the set $\Sigma_u$ of points $x$ where $\frac{1}{r^n}u \left (\frac{\cdot-x}{r} \right )$ converges, in the $L^1_{\mathrm{loc}}$-topology, to a non-constant function can be covered by a countable union of $(n-1)$-dimensional Lipschitz graphs.
\begin{itemize}
\item The results in \cite{delnin21} rely on the fact that the measure $\mu$ is induced by a $L^1_{\mathrm{loc}}$ function $u$ and that the rescalings converge in the $L^1_{\mathrm{loc}}$-topology, while in $\mathcal{S}_\mu$ the convergence is intended in the weak*-topology of Radon measures. Thus the subset of $\mathcal{S}_\mu$ where the unique blow-up of $u$ is not a constant function can be larger than $\Sigma_u$.
Is this set rectifiable as well? What can be said if $\mu$ is a general Radon measure?

\item It is easily seen (Lemma \ref{lmm:0-hom}) that the unique blow-up $\mu_x$ of $\mu$ at a point in $x \in \mathcal{S}_\mu$ has an \emph{invariant linear subspace} $D_x$ of $\mathbb{R}^n$, namely $(\tau_{y,1})_\# \mu_x = \mu_x$ for $y \in D_x$. Therefore
\begin{equation}\label{eq:def_s_mu_k}
\mathcal{S}_\mu = \bigcup_{k=1}^n \mathcal{S}_\mu^k,
\qquad\text{where} \qquad
\mathcal{S}_\mu^k \coloneqq \left \{x \in \mathcal{S}_\mu \colon \dim D_x=k \right \}.
\end{equation}
Do these $\mathcal{S}_\mu^k$ enjoy finer rectifiability properties?
\end{itemize}

The main result of this note provides an answer to these questions.

\begin{lemma}\label{lmm:0-hom}
Let $\mu$ be a signed Radon measure on $\mathbb{R}^n$ and let $\mu_x \neq 0$ be the unique blow-up of $\mu$ at $x \in \mathcal{S}_\mu$. Then there exists $\alpha=\alpha(x)\geq 0$ such that $\mu_x$ and $|\mu_x|$ are $(\alpha-n)$-homogeneous,\footnote{According to this definition, the measure induced by a $p$-homogenous locally integrable function $f$, with respect to the Lebesgue measure, is $p$-homogeneous as well.} namely
\begin{equation}\label{eq:hom_blow-up}
(\tau_{0,\lambda})_\# \mu_x=\lambda^\alpha \mu_x,
\quad
(\tau_{0,\lambda})_\# |\mu_x|=\lambda^\alpha |\mu_x|
\qquad
\forall \lambda>0.
\end{equation}
For $\mu_x$, and more generally for every $(\alpha-n)$-homogeneous Radon measure, the set
\begin{equation}
C_x\coloneqq \left \{ y \in \mathbb{R}^n \colon (\tau_{y,\lambda})_\# \mu_x=\lambda^\alpha (\tau_{y,1})_\# \mu_x \,\,\,\,\forall \lambda>0 \right \}
\end{equation}
coincides with the invariant subspace $D_x$ of $\mu_x$, that is
\begin{equation}
D_x \coloneqq \left \{ y \in \mathbb{R}^n \colon (\tau_{y,1})_\# \mu_x= \mu_x \right \}.
\end{equation}
Moreover $D_x$ is a linear subspace of $\mathbb{R}^n$ and, provided $\mu_x \neq 0$, it holds $\dim D_x \leq \alpha$.
\end{lemma}

This problem has already received some attention: in \cite{delnin21} it is proved that, for a given $u \in L^1_{\text{loc}}(\mathbb{R}^n)$, the set $\Sigma_u$ of points $x$ where $\frac{1}{r^n}u \left (\frac{\cdot-x}{r} \right )$ converges, in the $L^1_{\mathrm{loc}}$-topology, to a non-constant function can be covered by a countable union of $(n-1)$-dimensional Lipschitz graphs.
\begin{itemize}
\item The results in \cite{delnin21} rely on the fact that the measure $\mu$ is induced by a $L^1_{\mathrm{loc}}$ function $u$ and that the rescalings converge in the $L^1_{\mathrm{loc}}$-topology, while in $\mathcal{S}_\mu$ the convergence is intended in the weak*-topology of Radon measures. Thus the subset of $\mathcal{S}_\mu$ where the unique blow-up of $u$ is not a constant function can be larger than $\Sigma_u$.
Is this set rectifiable as well? What can be said if $\mu$ is a general Radon measure?

The main result of this note provides an answer to these questions.

The idea of the proof is that, for every $x \in \mathcal{S}_\mu^k$, there is a radius $r>0$ for which the points in $\mathcal{S}_\mu^k \cap B_r(x)$ with similar properties are contained in a cone $x+C$ around $x+D_x$. Otherwise, by contradiction, one can find a sequence of such points $y_j \notin x+C$ with $r_j=|x-y_j| \to 0$ and such that the measures $c_j \mu_{y_j}$ and $\mu_{y_j,r_j}$ are as close as we want. Now, on one hand $c_j\mu_{y_j}$ converges to a measure $\nu$ whose invariant subspace is close to $D_x$; on the other hand $\mu_{y_j,r_j}$, being translations of $\mu_{x,r_j}$ with respect to points outside $C$, converge to a translation of $\mu_x$ whose invariant subspace is not contained in $C$, contradicting the closeness of $c_j\mu_{y_j}$ and $\mu_{y_j,r_j}$.

\begin{corollario}\label{cor:rectif_criterion}
Let $\mu$ be a signed Radon measure on $\mathbb{R}^n$ and $k \in \{0,1,\dots,n\}$. Let us assume that, for $|\mu|$-a.e.\ $x \in \mathbb{R}^n$, there exists a Radon measure $\nu_x$ whose invariant linear subspace $V_x$ is $k$-dimensional and with the property that, for every sequence $r_j \to 0^+$ there are a subsequence, not relabeled, and a constant $c\neq 0$ such that
\begin{equation}
\frac{1}{r_j^k} (\tau_{x,r_j})_\# \mu
\overset{\ast}{\rightharpoonup} c \nu_x.
\end{equation}
Then $\mu$ is $k$-rectifiable: there exists a signed function $\theta \in L^1_{\mathrm{loc}}(\mathcal{S}_\mu^k,\mathcal{H}^k)$ such that \mbox{$\mu= \theta \mathcal{H}^k \llcorner \mathcal{S}_\mu^k$,} where $\mathcal{S}_\mu^k$ is defined in \eqref{eq:def_s_mu_k}.
\end{corollario}

\begin{corollario}\label{cor:almost_everywhere_rect}
Let $\mu$ be a signed Radon measure on $\mathbb{R}^n$ and $k \in \{0,1,\dots,n\}$. Then, for $|\mu|$-a.e.\ $x \in \mathcal{S}_\mu^k$, the unique blow-up of $\mu$ at $x$ is a multiple of $\mathcal{H}^k\llcorner D_x$.
\end{corollario}

\begin{lemma}\label{lmm:0-hom}
Let $\mu$ be a signed Radon measure on $\mathbb{R}^n$ and let $\mu_x \neq 0$ be the unique blow-up of $\mu$ at $x \in \mathcal{S}_\mu$. Then there exists $\alpha=\alpha(x)\geq 0$ such that $\mu_x$ and $|\mu_x|$ are $(\alpha-n)$-homogeneous,\footnote{According to this definition, the measure induced by a $p$-homogenous locally integrable function $f$, with respect to the Lebesgue measure, is $p$-homogeneous as well.} namely
\begin{equation}\label{eq:hom_blow-up}
(\tau_{0,\lambda})_\# \mu_x=\lambda^\alpha \mu_x,
\quad
(\tau_{0,\lambda})_\# |\mu_x|=\lambda^\alpha |\mu_x|
\qquad
\forall \lambda>0.
\end{equation}
For $\mu_x$, and more generally for every $(\alpha-n)$-homogeneous Radon measure, the set
\begin{equation}
C_x\coloneqq \left \{ y \in \mathbb{R}^n \colon (\tau_{y,\lambda})_\# \mu_x=\lambda^\alpha (\tau_{y,1})_\# \mu_x \,\,\,\,\forall \lambda>0 \right \}
\end{equation}
coincides with the invariant subspace $D_x$ of $\mu_x$, that is
\begin{equation}
D_x \coloneqq \left \{ y \in \mathbb{R}^n \colon (\tau_{y,1})_\# \mu_x= \mu_x \right \}.
\end{equation}
Moreover $D_x$ is a linear subspace of $\mathbb{R}^n$ and, provided $\mu_x \neq 0$, it holds $\dim D_x \leq \alpha$.
\end{lemma}

\begin{proof}[Proof of Theorem \ref{thm:main}]
The statement is trivial for $k=n$, thus we assume $k \in \{0,1,\dots,n-1\}$ be fixed throughout the proof.
Let us moreover fix $\gamma \in \left (0,\frac{1}{16}\right )$. It is possible to choose a finite number $m$ of $k$-dimensional linear subspaces $\{V_\ell\}_{\ell=1}^m$ such that every $k$-dimensional linear subspace $V$ is contained in $\overline{C(0,V_\ell,\gamma)}$ for some $\ell \in \{1,\dots,m\}$. For every $\ell \in \{1,\dots,m\}$ and every $i \in \mathbb{N}$, we define the set
\begin{equation}
E_{\ell,i} \coloneqq \left \{ x \in \mathcal{S}^k \colon D_x \subset \overline{C(0,V_\ell,\gamma)},\,\,\, \alpha(x) \leq i,\,\,\, \eta(x)\geq \frac{1}{i},\,\,\, d(c\mu_{x},\omega)\geq \frac{1}{i} \,\,\,\forall |c| \in \left [\frac{1}{i},1 \right ],\, \forall\omega \in \operatorname{Hom}_{k+1}^i \right \},
\end{equation}
where $D_x$ and $\alpha(x)$ are respectively the invariant subspace and the homogeneity exponent of $\mu_x$ given by Lemma \ref{lmm:0-hom}, $\eta=\eta(x)$ is given by Lemma \ref{lmm:lower_bound_mass_blow-up}, while $d$ and $\operatorname{Hom}_{k+1}^i$ are respectively the distance and the set of $(\alpha-n)$-homogeneous measures described in Section \ref{sec:notations}.
Since $\mathcal{S}^k= \bigcup_{i \in \mathbb{N}}\bigcup_{\ell=1}^m E_{i,\ell}$, it is enough to prove that each $E_{i,\ell}$ can be covered by a countable union of Lipschitz graphs.

Let us fix $i \in \mathbb{N}$ and $\ell \in \{1,\dots,m\}$.
Then fix the value $\varepsilon$ given by Lemma \ref{lmm:too_close} for $\beta=i$, $\delta=\frac{1}{i}$, $M=i \cdot 2^\alpha$ and, for every $\rho>0$, define the set
\begin{equation}
E_{i,\ell,\rho} \coloneqq
\left \{x \in E_{i,\ell} \colon \inf_{|c| \in [\eta,1]}d(\mu_{x,r},c\mu_x)< \varepsilon \,\,\,\forall r \in (0,\rho) \right \}.
\end{equation}
If $x \in E_{i,\ell}$, then it belongs to $E_{i,\ell,\rho}$ for some $\rho>0$, because the existence of a sequence $r_j \to 0$ such that $d(\mu_{x,r_j},c\mu_x) \geq \varepsilon$ for every $j \in \mathbb{N}$ and every $|c| \in [\eta,1]$ would contradict $x \in \mathcal{S}$. Therefore
\begin{equation}
E_{i,\ell} = \bigcup_{\rho \in \mathbb{Q} \cap (0,+\infty)} E_{i,\ell,\rho}
\end{equation}
and it suffice to prove that, for a fixed $\rho>0$, the set $E_{i,\ell,\rho}$ can be covered with a countable union of Lipschitz graphs.
In order to do so, we claim that, for every $x \in E_{i,\ell,\rho}$, there exists $r>0$ such that
\begin{equation}\label{eq:outside_cones}
E_{i,\ell,\rho} \cap B_{r}(x) \setminus C(x,D_x,3\gamma)= \{x\}.
\end{equation}
Let us fix $x\in E_{i,\ell,\rho}$, let $\mu_x$ be $(\alpha-n)$-homogeneous and let us assume, toward a contradiction, that for every $j \in \mathbb{N}$ there exists
\begin{equation}\label{eq:y_j_final}
y_j \in E_{i,\ell,\rho} \cap \overline{B_{\frac{r_j}{2}}(x)} \setminus\Big( B_{\frac{r_j}{4}}(x) \cup C(x,D_x,3\gamma) \Big),
\end{equation}
for some $r_j \leq \rho$ such that $r_j \downarrow 0$. This in particular implies the existence of constants $c_j$ with $|c_j| \in \left [\frac{1}{i},1 \right ]$ such that
\begin{equation}\label{eq:distance_muj}
d( c_j\mu_{y_j}, \mu_{y_j,r_j})< \varepsilon
\qquad
\forall j \in \mathbb{N}.
\end{equation}
Up to selecting a non relabeled subsequence and without loss of generality, we can assume $c_j>0$ and that $\mu_{x,r_j} \overset{\ast}{\rightharpoonup} c\mu_x$ for some $c \in \left [\frac{1}{i},1 \right ]$.
We now analyze the weak*-limit (up to a suitable subsequence) of the two sequences of measures in \eqref{eq:distance_muj}.

The idea of the proof is that, for every $x \in \mathcal{S}_\mu^k$, there is a radius $r>0$ for which the points in $\mathcal{S}_\mu^k \cap B_r(x)$ with similar properties are contained in a cone $x+C$ around $x+D_x$. Otherwise, by contradiction, one can find a sequence of such points $y_j \notin x+C$ with $r_j=|x-y_j| \to 0$ and such that the measures $c_j \mu_{y_j}$ and $\mu_{y_j,r_j}$ are as close as we want. Now, on one hand $c_j\mu_{y_j}$ converges to a measure $\nu$ whose invariant subspace is close to $D_x$; on the other hand $\mu_{y_j,r_j}$, being translations of $\mu_{x,r_j}$ with respect to points outside $C$, converge to a translation of $\mu_x$ whose invariant subspace is not contained in $C$, contradicting the closeness of $c_j\mu_{y_j}$ and $\mu_{y_j,r_j}$.

\medskip

A first outcome of Theorem \ref{thm:main} is a rectifiability criterion for signed Radon measures in terms of their blow-ups which, to the best of our knowledge, is new.

\begin{corollario}\label{cor:rectif_criterion}
Let $\mu$ be a signed Radon measure on $\mathbb{R}^n$ and $k \in \{0,1,\dots,n\}$. Let us assume that, for $|\mu|$-a.e.\ $x \in \mathbb{R}^n$, there exists a Radon measure $\nu_x$ whose invariant linear subspace $V_x$ is $k$-dimensional and with the property that, for every sequence $r_j \to 0^+$ there are a subsequence, not relabeled, and a constant $c\neq 0$ such that
\begin{equation}
\frac{1}{r_j^k}  (\tau_{x,r_j})_\# \mu
\overset{\ast}{\rightharpoonup} c \nu_x.
\end{equation}
Then $\mu$ is $k$-rectifiable: there exists a signed function $\theta \in L^1_{\mathrm{loc}}(\mathcal{S}_\mu^k,\mathcal{H}^k)$ such that \mbox{$\mu= \theta \mathcal{H}^k \llcorner \mathcal{S}_\mu^k$,} where $\mathcal{S}_\mu^k$ is defined in \eqref{eq:def_s_mu_k}.
\end{corollario}

The above result extends to signed Radon measures the well-known criterion for positive ones, see for instance \cite[Theorem 11.8]{simonlectures83}, where one usually sets $\nu_x=\theta_0 \mathcal{H}^k \llcorner V_x$ for some $k$-dimensional linear subspace $V_x$ and assumes the existence of the limit $r^{-k}(\tau_{x,r_j})_\# \mu \overset{\ast}{\rightharpoonup} \theta_0 \mathcal{H}^k \llcorner V_x$ as $r \to 0^+$.

We point out that Corollary \ref{cor:rectif_criterion} cannot be trivially deduced from the corresponding result for positive measures, since the fact that $\mu$ has a certain blow-up at a point does not provide any suitable information on the blow-ups of $|\mu|$ at the same point, not even their uniqueness; nor does it appear that the classical proof of \cite[Theorem 11.8]{simonlectures83} can be readily adapted to the case of signed measures.

\medskip

A further consequence of Theorem \ref{thm:main} is the following extension of \cite[Theorem 3.2]{mattila_unique05} to signed Radon measures. It is based on the intuitive idea that, since each $\mathcal{S}_{\mu}^k$ is $k$-rectifiable, it is reasonable to expect that its subset of points where the blow-up of $\mu$ is more diffused than a $k$-dimensional measure is $|\mu|$-negligible.

\begin{corollario}\label{cor:almost_everywhere_rect}
Let $\mu$ be a signed Radon measure on $\mathbb{R}^n$ and $k \in \{0,1,\dots,n\}$. Then, for $|\mu|$-a.e.\ $x \in \mathcal{S}_\mu^k$, the unique blow-up of $\mu$ at $x$ is a multiple of $\mathcal{H}^k\llcorner D_x$.
\end{corollario}

We conclude this introduction with a comment on the definition of unique blow-up at a point.
As already said above, we say that $\mu_x$ is the unique blow-up of $\mu$ at $x \in \operatorname{supp}\mu$ if, for every sequence $r_j \to 0$, \emph{there exists} a subsequence for which \eqref{eq:def-unique_blowup} holds true. This differs from the definition used in \cite{mattila_unique05} for positive Radon measures, where one does not require the convergence of a subsequence for every choice of $\{r_j\}_j$: according to \cite{mattila_unique05}, one says that  $\mu$ has unique blow-up at $x \in \mathbb{R}$ if there exists $\nu$ such that any non-trivial weak*-limit (in case it exists) of $c_j (\tau_{x,r_j})_\# \mu$, where $c_j>0$ and $r_j \to 0^+$, is of the form $c\nu$ for some $c>0$.

\cite[Lemma 2.5]{mattila_unique05} shows, provided $\mu$ is positive, that this condition actually implies $\mu_{x,r} \overset{\ast}{\rightharpoonup} c\mu_x$ as $r \to 0$ and, subsequently, our definition (they are actually equivalent).
However, this does not hold in the case of signed Radon measures: let us consider two sequences of positive real numbers $a_j, \rho_j$ such that
\begin{equation}
\lim_{j \to +\infty} \frac{1}{a_j} \sum_{i=j+1}^{+\infty} a_i = 0,
\qquad
\lim_{j \to +\infty} \frac{\rho_{j+1}}{\rho_j}=0.
\end{equation}
Then the Radon measure on $\mathbb{R}$ defined as
\begin{equation}
\mu \coloneqq \sum_{j \in \mathbb{N}} a_j\left ( \delta_{2\rho_j + \rho_j^2} - \delta_{2\rho_j - \rho_j^2} \right )
\end{equation}
has the null measure as unique blow-up at $x=0$ in the sense of \cite{mattila_unique05}, but $\mu_{0,\rho_j} =\frac{1}{|\mu|(B_{\rho_j}(0))} (\tau_{0,\rho_j})_\# \mu $ has no converging subsequences because $|\mu_{x,\rho_j}|\big(B_2(0)\big) \to +\infty$ as $j\to +\infty$.

\subsection*{Acknowledgments}
The author would like to thank Andrea Marchese, Paolo Bonicatto and Gian Paolo Leonardi for inspiring conversations. He has been supportend by PRIN project 2022PJ9EFL ``Geometric Measure Theory: Structure of Singular Measures, Regularity Theory and Applications in the Calculus of Variations" CUP:E53D23005860006 and  partially supported by INDAM-GNAMPA. Part of the work has been carried out during the thematic program on ``Free boundary problems" at ESI (Wien), which the author wishes to thank.

The post-theorem introduction gives the following proof idea for Theorem~\ref{thm:main}.

1. **Cone containment near each point.** For every $x\in\mathcal{S}_\mu^k$, there exists $r>0$ such that the points in $\mathcal{S}_\mu^k\cap B_r(x)$ with similar properties are contained in a cone $x+C$ around $x+D_x$.

2. **Contradiction setup if cone containment fails.** Otherwise, one can find a sequence $y_j\notin x+C$ with $r_j=|x-y_j|\to 0$ such that the measures $c_j\mu_{y_j}$ and $\mu_{y_j,r_j}$ are “as close as we want.”

3. **Two limiting descriptions of the same asymptotically close measures.**
   - On the one hand, $c_j\mu_{y_j}$ converges to a measure $\nu$ whose invariant subspace is close to $D_x$.
   - On the other hand, $\mu_{y_j,r_j}$ are translations of $\mu_{x,r_j}$ “with respect to points outside $C$,” hence they converge to a translation of $\mu_x$ whose invariant subspace is not contained in $C$.

4. **Contradiction.** This “contradict[s] the closeness of $c_j\mu_{y_j}$ and $\mu_{y_j,r_j}$,” forcing the desired cone property; this yields the Lipschitz-graph covering and the identification of the approximate tangent plane with $D_x$ for $\mathcal{H}^k$-a.e. $x\in\mathcal{S}_\mu^k$.

The post-theorem introduction gives the following proof idea for the main theorem.

1. **Cone containment near each point.** For every $x\in\mathcal{S}_\mu^k$, there exists $r>0$ such that the points in $\mathcal{S}_\mu^k\cap B_r(x)$ with similar properties are contained in a cone $x+C$ around $x+D_x$.

2. **Contradiction setup if cone containment fails.** Otherwise, one can find a sequence $y_j\notin x+C$ with $r_j=|x-y_j|\to 0$ such that the measures $c_j\mu_{y_j}$ and $\mu_{y_j,r_j}$ are “as close as we want.”

3. **Two limiting descriptions of the same asymptotically close measures.**
   - On the one hand, $c_j\mu_{y_j}$ converges to a measure $\nu$ whose invariant subspace is close to $D_x$.
   - On the other hand, $\mu_{y_j,r_j}$ are translations of $\mu_{x,r_j}$ “with respect to points outside $C$,” hence they converge to a translation of $\mu_x$ whose invariant subspace is not contained in $C$.

4. **Contradiction.** This “contradict[s] the closeness of $c_j\mu_{y_j}$ and $\mu_{y_j,r_j}$,” forcing the desired cone property; this yields the Lipschitz-graph covering and the identification of the approximate tangent plane with $D_x$ for $\mathcal{H}^k$-a.e. $x\in\mathcal{S}_\mu^k$.

\label{thm:main}
Let $\mu$ be a signed Radon measure on $\mathbb{R}^n$ and $k \in \{0,1,\dots,n\}$. Then $\mathcal{S}_\mu^k$ is contained in a countable union of $k$-dimensional Lipschitz graphs. Moreover, the approximate tangent plane to $\mathcal{S}_\mu^k$ at $x$ coincides with $D_x$ for $\mathcal{H}^{k}$-a.e.\ $x \in \mathcal{S}_\mu^k$.,

Question:
Let \(\mu\) be a signed Radon measure on \(\mathbb{R}^n\). For \(x\in \operatorname{supp}\mu\), say that \(\mu\) has a unique blow-up at \(x\) if there exists a Radon measure \(\mu_x\), with either \(|\mu_x|(B_1(0))=1\) or \(\mu_x=0\), such that for every sequence \(r_j\to 0^+\) there are a subsequence and a constant \(c\neq 0\) for which
\[
\mu_{x,r_j}:=\frac{1}{|\mu|(B_{r_j}(x))}(\tau_{x,r_j})_\#\mu \overset{*}{\rightharpoonup} c\mu_x,
\qquad \tau_{x,r}(y)=\frac{y-x}{r}.
\]
Let \(\mathcal S_\mu\) be the set of points where \(\mu\) has a unique blow-up. For \(x\in \mathcal S_\mu\), let
\[
D_x:=\{y\in\mathbb R^n:(\tau_{y,1})_\#\mu_x=\mu_x\}
\]
be the invariant linear subspace of the blow-up, and for \(k\in\{0,1,\dots,n\}\) define
\[
\mathcal S_\mu^k:=\{x\in\mathcal S_\mu:\dim D_x=k\}.
\]
Which statement holds for every signed Radon measure \(\mu\) on \(\mathbb R^n\) and every \(k\in\{0,1,\dots,n\}\)?

The set \(\mathcal S_\mu^k\) is contained in a countable union of \(k\)-dimensional Lipschitz graphs, and for \(\mathcal H^k\)-almost every \(x\in \mathcal S_\mu^k\), the approximate tangent plane to \(\mathcal S_\mu^k\) at \(x\) is exactly \(D_x\).

Choices:
B. The set \(\mathcal S_\mu^k\) is contained in a countable union of \((k+1)\)-dimensional Lipschitz graphs, and for \(\mathcal H^{k+1}\)-almost every \(x\in \mathcal S_\mu^k\), the approximate tangent plane to \(\mathcal S_\mu^k\) at \(x\) contains \(D_x\).
C. The set \(\mathcal S_\mu^k\) is contained in a countable union of \(k\)-dimensional Lipschitz graphs.
D. The set \(\mathcal S_\mu^k\) is contained in a countable union of \(k\)-dimensional Lipschitz graphs, and for every \(x\in \mathcal S_\mu^k\), the approximate tangent plane to \(\mathcal S_\mu^k\) at \(x\) exists and is exactly \(D_x\).
E. The set \(\mathcal S_\mu^k\) is contained in a finite union of \(k\)-dimensional Lipschitz graphs, and for \(\mathcal H^k\)-almost every \(x\in \mathcal S_\mu^k\), the approximate tangent plane to \(\mathcal S_\mu^k\) at \(x\) is a \(k\)-plane parallel to \(D_x\).


Question:
Let $\mu$ be a signed Radon measure on $\mathbb{R}^n$, and for $x\in \operatorname{supp}\mu$ and $r>0$ define the rescaled measure
\[
\mu_{x,r}:=\frac{1}{|\mu|(B_r(x))}(\tau_{x,r})_\#\mu,
\qquad \tau_{x,r}(y):=\frac{y-x}{r}.
\]
Say that $\mu$ has a unique blow-up at $x$ if there exists a Radon measure $\mu_x$ with either $|\mu_x|(B_1(0))=1$ or $\mu_x=0$ such that, for every sequence $r_j\to 0^+$, one can extract a subsequence and find a constant $c\neq 0$ for which
\[
\mu_{x,r_j}\overset{*}{\rightharpoonup} c\mu_x
\]
in the weak-* topology of Radon measures. Let $\mathcal S_\mu$ be the set of points where $\mu$ has unique blow-up. For $x\in \mathcal S_\mu$, define the invariant subspace of the blow-up by
\[
D_x:=\{y\in \mathbb R^n:(\tau_{y,1})_\#\mu_x=\mu_x\},
\]
and, for $k\in\{0,1,\dots,n\}$, set
\[
\mathcal S_\mu^k:=\{x\in \mathcal S_\mu:\dim D_x=k\}.
\]
Under these assumptions, which statement about $\mathcal S_\mu^k$ holds?

The set $\mathcal S_\mu^k$ is contained in a countable union of $k$-dimensional Lipschitz graphs. Moreover, for $\mathcal H^k$-almost every $x\in \mathcal S_\mu^k$, the approximate tangent plane to $\mathcal S_\mu^k$ at $x$ exists and coincides with $D_x$.

Choices:
B. The set $\mathcal S_\mu^k$ is contained in a countable union of $(k+1)$-dimensional Lipschitz graphs. Moreover, for $\mathcal H^{k+1}$-almost every $x\in \mathcal S_\mu^k$, the approximate tangent plane to $\mathcal S_\mu^k$ at $x$ exists and contains $D_x$.
C. The set $\mathcal S_\mu^k$ is countably $\mathcal H^k$-rectifiable; in particular, it is contained in a countable union of Lipschitz images of subsets of $\mathbb R^k$.
D. The set $\mathcal S_\mu^k$ is contained in a countable union of $k$-dimensional Lipschitz graphs. Moreover, for every $x\in \mathcal S_\mu^k$, the approximate tangent plane to $\mathcal S_\mu^k$ at $x$ exists and coincides with $D_x$.
E. The set $\mathcal S_\mu^k$ is contained in a finite union of $k$-dimensional Lipschitz graphs. Moreover, for $\mathcal H^k$-almost every $x\in \mathcal S_\mu^k$, the approximate tangent plane to $\mathcal S_\mu^k$ at $x$ exists and is parallel to $D_x$.