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# Some examples of separable convex-cocompact subgroups

Mark Hagen  School of Mathematics, University of Bristol, Bristol, UK
<EMAIL_ADDRESS>and  Alessandro Sisto  Maxwell Institute and Department
of Mathematics, Heriot-Watt University, Edinburgh, UK<EMAIL_ADDRESS>
###### Abstract.

Reid asked whether all convex-cocompact subgroups of mapping class groups are
separable. Using a construction of Manning-Mj-Sageev, we give examples of
separable convex-cocompact subgroups that are free of arbitrary finite rank,
while prior examples seem to all be virtually cyclic.

##  1\. Introduction

A subgroup $H$ of a group $G$ is _separable (in $G$)_ if, for all $g\notin H$,
there exists a finite group $F$ and a homomorphism $\phi:G\to F$ such that
$\phi(g)\notin\phi(H)$. Separability arises naturally in topology when, given
an immersed submanifold $N$ of a manifold $M$, one tries to find a finite
cover of $M$ where $N$ embeds; see [Sco78]. Notably, separability, and more
specifically separability of quasiconvex subgroups of certain hyperbolic
groups, played a crucial role in the resolution of the Virtual Haken and
Virtual Fibering Conjectures [Wis21, Ago13].

In the context of mapping class groups, the closest analogues of quasiconvex
subgroups of hyperbolic groups are _convex-cocompact_ subgroups, as defined in
[FM02]; see Subsection 2.1 for the characterisation that we will use. In
analogy with the case of hyperbolic manifolds, Reid asked whether convex-
cocompact subgroups of mapping class groups are separable [Rei06, Question
3.5]. As far as we are aware, the only examples of convex-cocompact subgroups
known to be separable are virtually cyclic, as covered by [LM07]. However, it
is shown in [BHMS20] that if "enough" hyperbolic groups are residually finite
then all convex-cocompact subgroups are separable. (Note that it is known that
not all finitely generated subgroups of mapping class groups are separable
[LM07].)

The main goal of this paper is to provide the first examples of separable
convex-cocompact subgroups that are not virtually cyclic, using the criterion
in Theorem 1.1.

Write $\Sigma_{\mathfrak{g}}$ to mean the closed connected orientable surface
of genus $\mathfrak{g}$ and $MCG(\Sigma_{\mathfrak{g}})$ its mapping class
group. Given $H\leq MCG(\Sigma_{\mathfrak{g}})$, denote by $\Gamma_{H}$ the
corresponding extension of $H$. This can be seen as the preimage of $H$ under
the natural map $MCG(\Sigma_{\mathfrak{g}}-\\{p\\})\to
MCG(\Sigma_{\mathfrak{g}})$ occurring in the Birman exact sequence (see e.g.
[FM02, Section 1.2] for detailed description of $\Gamma_{H}$). When $H$ is
convex-cocompact, $\Gamma_{H}$ is hyperbolic [Ham05] (if in addition $H$ is
free, as it will be in our examples, hyperbolicity of $\Gamma_{H}$ is also
[FM02, Theorem 1.3]).

The criterion is:

######  Theorem 1.1.

Let $\mathfrak{g}\geq 1$ and let $H$ be a torsion-free malnormal convex-
cocompact subgroup of $MCG(\Sigma_{\mathfrak{g}})$. If $\Gamma_{H}$ is
conjugacy separable, then $H$ is separable in $MCG(\Sigma_{\mathfrak{g}})$.

To apply the criterion, we need examples of convex-cocompact subgroups $H$
with $\Gamma_{H}$ conjugacy separable. These come from work of Manning-Mj-
Sageev [MMS19] – who gave examples where $\Gamma_{H}$ is virtually compact
special – combined with Minasyan–Zalesskii’s result on conjugacy separability
of hyperbolic virtually special groups [MZ16]. So, the examples provided by
the following theorem come from [MMS19]; we just observe that their
construction gives rise to malnormal subgroups in many cases.

######  Theorem 1.2.

For every $\mathfrak{g}\geq 3$ and $n\geq 1$, there exist infinitely many
conjugacy classes of malnormal convex-cocompact subgroups
$H<MCG(\Sigma_{\mathfrak{g}})$ such that $H$ is isomorphic to a free group of
rank $n$ whose corresponding extension $\Gamma_{H}$ is virtually compact
special.

Recalling that convex-cocompactness of $H$ implies hyperbolicity of
$\Gamma_{H}$, combining the above theorem with [MZ16, Theorem 1.1] and Theorem
1.1 gives:

######  Corollary 1.3.

For every $\mathfrak{g}\geq 3$ and $n\geq 1$ there exist infinitely many
conjugacy classes of separable convex-cocompact subgroups $H$ of
$MCG(\Sigma_{\mathfrak{g}})$, with each $H\cong F_{n}$.

We note that Theorem 1.1 does not apply to arbitrary cyclic convex-cocompact
subgroups, even though, in this case, $\Gamma_{H}$ is the fundamental group of
a fibred hyperbolic 3-manifold, which is virtually compact special [Duf12,
Ago13]. Indeed, such cyclic subgroups may not be malnormal, and the maximal
elementary subgroups containing them may not be torsion-free (in case the
maximal elementary subgroup containing the cyclic group is torsion-free, the
criterion does apply, though). We wonder:

######  Question 1.4.

To what extent can malnormality and/or torsion-freeness be relaxed in Theorem
1.1?

### Outline of proofs

In this paper, we did not pursue the most general versions of our intermediate
results, instead leaving questions that we believe to be of independent
interest; we highlight the main ones below.

To prove Theorem 1.1, we use a version of Grossman’s criterion for residual
finiteness of an outer automorphism group [Gro75] (Grossman’s criterion is one
way to prove residual finiteness of $MCG(\Sigma_{\mathfrak{g}})$). This is
Proposition 2.5 and has two hypotheses. The first one, in the case of convex-
cocompact subgroups, is conjugacy separability of the extension group. The
second one translates to a question about automorphisms of surface groups.
Roughly, given a surface group $G$, a group $K$ of automorphisms of $G$, and
automorphism $\phi$ not in $K$, one can ask whether there exists $x\in G$ such
that $\phi(x)$ is not conjugate to $k(x)$ for any $k\in K$. In our case, we
are interested in $K=\Gamma_{H}$ for some convex-cocompact subgroup $H$ (where
the mapping class group of the punctured surface can be identified with an
index 2 subgroup of the automorphism group of $G$).

We check this second condition for suitable $H$ in two steps. First,
Proposition 3.1 reduces the problem above to an analogous problem where $K$ is
replaced by finitely many automorphisms. This is done by showing that left
cosets of $H$ contain finitely many elements that are not pseudo-Anosov. This
is where we use the hypotheses that $H$ is torsion-free and malnormal.

The second step is to construct a suitable $x$, given finitely many
automorphisms. We do so specifically for surface groups in Proposition 4.1,
but, inspired by the fact that Grossman’s criterion can be applied to
acylindrically hyperbolic groups [AMS16], we ask:

######  Question 1.5.

Let $G$ be a torsion-free acylindrically hyperbolic group, and let
$\phi_{1},\dots,\phi_{n}$ be non-inner automorphisms of $G$. Does there exist
$x\in G$ with $x$ and $\phi_{i}(x)$ non-conjugate for all $i$?

In fact, we do not know the answer to this question even in the special case
where $G$ is hyperbolic. One way of answering the question affirmatively would
be to show that a "generic" $x$ has the required property, which would be
interesting in its own right. That is:

######  Question 1.6.

Let $G$ be a torsion-free acylindrically hyperbolic group, let $\phi$ be a
non-inner automorphism of $G$, and let $(w_{n})$ be a simple random walk on
$G$. Is it true that, with probability going to $1$ as $n$ goes to infinity,
$w_{n}$ is not conjugate to $\phi(w_{n})$?

This question is already of interest for $G$ hyperbolic and the techniques of
[MS19] seem relevant.

### Acknowlegdments

We are grateful to Michah Sageev for answering a question about MMS subgroups
and to Ashot Minasyan for several very helpful comments and corrections. We
also thank the referee for a number of suggestions that improved the paper.

##  2\. Preliminaries

###  2.1. Curve graphs and convex-cocompactness

Given a finite-type surface $\Sigma$, we denote by $\mathcal{C}(\Sigma)$ its
curve graph. For all finite-type surfaces $\Sigma$, the curve graph
$\mathcal{C}(\Sigma)$ is hyperbolic [MM99] and, in fact, say, $100$-hyperbolic
[HPW15], see also [Aou13, Bow14, PS17]. Also, the action of $MCG(\Sigma)$ on
$\mathcal{C}(\Sigma)$ is acylindrical [Bow08]. A finitely generated subgroup
of $MCG(\Sigma)$ is _convex-cocompact_ if the orbit maps to
$\mathcal{C}(\Sigma)$ are quasi-isometric embeddings [KL08, Theorem 1.3].

###  2.2. Elementary subgroups

Throughout the paper, we use the following notation. If $X$ is a hyperbolic
space and $Y$ is a quasiconvex subspace, then the closest-point projection
$\pi_{Y}$ is the bounded set defined by

$\pi_{Y}(x)=\\{y\in Y:d(x,y)\leq d(x,Y)+1\\}.$

If a group $A$ acts by isometries on $X$, then $\pi_{Y}$ is
$\operatorname{Stab}_{A}(Y)$–equivariant. Next, we summarise properties of
_elementary closures_ of loxodromic elements used later, as first considered
by [BF02].

######  Proposition 2.1.

Let the group $G$ act acylindrically on the hyperbolic space $X$. There exists
a constant $Q$ such that the following holds. Let $g\in G$ be a loxodromic
element. Then:

  1. (1)

There exists a unique maximal elementary subgroup $E(g)$ containing $g$, and a
$Q$-quasiconvex $E(g)$–invariant $Q-$quasiline $Y=Y_{g}$.

  2. (2)

For every $D\geq 0$ there exists $L\geq 0$ (depending on $D$ and $g$), such
that the following holds. If $h\in G$ has the property that there exist
$x,y\in Y$ with $d(x,y)\geq L$ and $d(x,h\cdot x),d(y,h\cdot y)\leq D$, then
$h\in E(g)$.

###### Proof.

Item (1) is given by [DGO17, Lemma 6.5]. Item (2) is [BF02, Proposition
6.(2)]. ∎

###  2.3. Intersection of conjugates vs coarse intersections of coset orbits

We also need a certain correspondence between the behaviour of orbits in
$\mathcal{C}(\Sigma)$ of cosets of convex-cocompact subgroups and
intersections of conjugates. The next lemma, which is used in the proof of
Proposition 3.1 below, follows from arguments in [AMST19]. We note that [AM21,
Proposition 4.1] is related to the lemma, though neither implies the other.

######  Lemma 2.2.

Let $\mathfrak{g}\geq 2$ and let $H$ be a convex-compact subgroup of
$MCG(\Sigma_{\mathfrak{g}})$. Let $o\in\mathcal{C}(\Sigma_{\mathfrak{g}})$.
Suppose that for some $x\in MCG(\Sigma_{\mathfrak{g}})$ and some $C\geq 0$,
the diameter of $xH\cdot o\cap N_{C}(H\cdot
o)\subseteq\mathcal{C}(\Sigma_{\mathfrak{g}})$ is infinite. Then the subgroup
$H\cap xHx^{-1}$ is infinite.

###### Proof.

The convex-cocompact subgroup $H$ in the statement, acting on
$\mathcal{C}(\Sigma_{\mathfrak{g}})$, satisfies an appropriately modified
version of the _WPD condition_ from [AMST19, Definition 2.12]. More
specifically, said definition is stated for the action on a Cayley graph with
respect to a possibly infinite generating sets. The equivariant quasi-isometry
classes of said Cayley graphs are in natural bijection with geodesic spaces
with a cobounded action, so dealing with Cayley graphs or cobounded actions is
equivalent.

The proof of [AMST19, Lemma 5.2], and hence of [AMST19, Proposition 5.3], only
uses the WPD property and hence it goes through in our setting (replacing
subgroups/cosets with their orbits in $\mathcal{C}(\Sigma_{\mathfrak{g}})$
etc.). In particular, we conclude that $xH\cdot o\cap N_{C}(H\cdot
o)\subseteq\mathcal{C}(\Sigma_{\mathfrak{g}})$ lies within finite Hausdorff
distance of $(H\cap xHx^{-1})\cdot o$. Since the former has infinite diameter,
we have that $H\cap xHx^{-1}$ is infinite, as required. ∎

###  2.4. Concatenating geodesics in hyperbolic spaces

Later, we will verify that certain elements are pseudo-Anosov by constructing
quasigeodesic axes using the following lemma, which we prove by reducing it to
[HW16, Lemma 4.2].

######  Lemma 2.3.

For every $\delta\geq 0$ there exists a constant $C_{\delta}$ such that the
following holds. Let $(\alpha_{i})_{i\in\mathbb{Z}}$ be geodesics in a
$\delta$-hyperbolic geodesic space, of length bounded independently of $i$,
with the terminal point of $\alpha_{i}$ being the initial point of
$\alpha_{i+1}$. Let

$d_{i,j}=diam(N_{3\delta}(\alpha_{i})\cap\alpha_{j})$

and assume that for all even $i$,

$\ell(\alpha_{i})\geq d_{i,i+1}+d_{i,i+2}+d_{i,i-1}+d_{i,i-2}+C_{\delta}.$

Then the concatenation of the $\alpha_{i}$ is a quasi-geodesic.

Figure 1. An illustration of the hypotheses in Lemma 2.3. The long horizontal
segment is $\alpha_{i}$ and the central subsegment is $\alpha_{i}^{\prime}$.

###### Proof.

We can assume $\delta>0$ for convenience. For $i$ even, let
$\alpha^{\prime}_{i}$ be the subgeodesic of $\alpha_{i}$ obtained by removing
the initial open segment of length $d_{i,i-1}+d_{i,i-2}$ and the final open
segment of length $d_{i,i+1}+d_{i,i+2}$.

For $i$ odd, let $\alpha^{\prime}_{i}$ be a geodesic joining the terminal
point of $\alpha^{\prime}_{i-1}$ to the initial point of
$\alpha^{\prime}_{i+1}$. So, the geodesics $\alpha^{\prime}_{i}$ concatenate
to form a bi-infinite path.

Considering thin quadrangles, we now show that that [HW16, Lemma 4.2] applies,
that is, we show that for $i$ even, the intersections

$N_{3\delta}(\alpha^{\prime}_{i})\cap\alpha^{\prime}_{i\pm 1}$

and

$N_{3\delta}(\alpha^{\prime}_{i})\cap\alpha^{\prime}_{i\pm 2}$

each have diameter at most $10\delta$.

We will do the argument for $\alpha^{\prime}_{i+1}$ and
$\alpha^{\prime}_{i+2}$, the other arguments being analogous. Denoting by
$\alpha^{\prime\prime}_{i}$ the terminal segment of $\alpha_{i}$ that starts
at the terminal point of $\alpha^{\prime}_{i}$, note that
$\alpha^{\prime}_{i+1}$ is contained in the $2\delta$-neighborhood of
$\alpha^{\prime\prime}_{i}\cup\alpha_{i+1}\cup\alpha_{i+2}$, and clearly so is
$\alpha^{\prime}_{i+2}$. Hence, it suffices to show that there is no point $p$
on $\alpha^{\prime}_{i}$ further than $20\delta$ from the terminal point of
$\alpha^{\prime}_{i}$ but within $5\delta$ of $\alpha_{i+1}\cup\alpha_{i+2}$
(there is clearly no point $p$ on $\alpha^{\prime}_{i}$ further than
$20\delta$ from the terminal point and within $5\delta$ of
$\alpha^{\prime\prime}_{i}$).

Figure 2. Proof of the nonexistence of $p$ in Lemma 2.3. Note that
$d(a_{i},p)\geq 20\delta$.

Suppose by contradiction that such $p$ exists, and in fact that it lies within
$5\delta$ of $\alpha_{i+2}$, the other case being easier. The following
argument is illustrated in Figure 2. Let $a_{i}$ be the terminal point of
$\alpha_{i}^{\prime}$, and let $\beta_{i}$ be the union of the terminal
segment of $\alpha_{i}^{\prime}$ of length $d(a_{i},p)-8\delta$, and
$\alpha_{i}^{\prime\prime}$.

Consider the $2\delta$–thin quadrilateral formed by the following four
geodesics:

  * •

a geodesic $\sigma$ of length at most $5\delta$ from $p$ to a point on
$\alpha_{i+2}$;

  * •

the subpath of $\alpha_{i+2}$ subtended by the terminal point of $\sigma$ and
$\alpha_{i+1}\cap\alpha_{i+2}$;

  * •

$\alpha_{i+1}$;

  * •

the terminal subpath of $\alpha_{i}$ from $p$ to the initial point of
$\alpha_{i+1}$.

By $2\delta$–thinness of the quadrilateral, the fourth path on the list is
contained in the union of the $2\delta$–neighbourhoods of the other three
paths. Since the initial point of $\beta_{i}$ is $8\delta$–far from $p$, and
$|\sigma|\leq 5\delta$, we see that no point of $\beta_{i}$ is $2\delta$–close
to $\sigma$, for otherwise it would be $7\delta$–close to $p$, a
contradiction. Hence any point on $\beta_{i}$ lies $2\delta$-close to either
$\alpha_{i+1}$ or $\alpha_{i+2}$.

Note that $\beta_{i}\cap N_{2\delta}(\alpha_{i+1})$ has diameter at most
$d_{i,i+1}+4\delta$. Indeed, if $p,q\in\beta_{i}$ are respectively
$2\delta$–close to points $p^{\prime},q^{\prime}\in\alpha_{i+1}$, then
$p^{\prime},q^{\prime}$ are in $N_{3\delta}(\alpha_{i})\cap\alpha_{i+1}$, so
$d(p^{\prime},q^{\prime})\leq d_{i,i+1}$, and hence, by the triangle
inequality, $d(p,q)\leq d_{i,i+1}+4\delta$. Similarly, $\beta_{i}\cap
N_{2\delta}(\alpha_{i+2})$ has diameter at most $d_{i,i+2}+4\delta$. Since
these two intersections cover $\beta_{i}$, we conclude that $|\beta_{i}|\leq
d_{i,i+1}+d_{i,i+2}+8\delta$.

On the other hand, $\beta_{i}$ is the concatenation of a terminal segment of
$\alpha_{i}^{\prime}$ of length at least $12\delta$ (since $p$ is assumed to
lie $20\delta$–far from the terminal point of $\alpha_{i}^{\prime}$) with the
path $\alpha_{i}^{\prime\prime}$ (whose length is $d_{i,i+1}+d_{i,i+2}$).
Hence $|\beta_{i}|>d_{i,i+1}+d_{i,i+2}+8\delta$. This is a contradiction, so
the point $p$ cannot exist.

Hence, for sufficiently large $C_{\delta}$, [HW16, Lemma 4.2] applies,
yielding that the concatenation of the $\alpha^{\prime}_{i}$ is a quasi-
geodesic. Hence, so is the concatenation of the $\alpha_{i}$, since all
geodesics involved have uniformly bounded length. ∎

###  2.5. Grossman criterion

We now give the generalisation of Grossman’s residual finiteness criterion
mentioned in the introduction. Assume that the group $G$ has trivial centre,
and let $i:G\to\mathrm{Inn}(G)$ be the natural isomorphism defined by
$i(g)(x)=gxg^{-1}$.

######  Remark 2.4.

For any $x\in G$ and $\gamma\in\operatorname{Aut}(G)$,

$\gamma i(x)\gamma^{-1}=i(\gamma(x)),$

just by expanding the definition of $i$.

We also need the usual exact sequence

$1\to\mathrm{Inn}(G)\to\operatorname{Aut}(G)\stackrel{{\scriptstyle\Psi}}{{\longrightarrow}}\mathrm{Out}(G)\to
1.$

The separability version of Grossman’s criterion is:

######  Proposition 2.5.

Let $G$ be a finitely generated group with trivial centre and let
$H<\mathrm{Out}(G)$. Let $\Gamma_{H}=\Psi^{-1}(H)$ and let
$\phi\in\operatorname{Aut}(G)$. Suppose that:

  1. (1)

there exists $x\in G$ such that $\phi(x)\neq h(x)$ for all $h\in\Gamma_{H}$.

  2. (2)

$\Gamma_{H}$ is conjugacy separable.

Then there exists a finite group $F$ and a homomorphism $q:\mathrm{Out}(G)\to
F$ with $q(\Psi(\phi))\notin q(H)$.

######  Remark 2.6.

Taking $H=\\{1\\}$ in the above proposition yields [Gro75, Theorem 1] in the
case where $G$ has trivial centre.

###### Proof of Proposition 2.5.

Item (1) provides $x\in G$ such that $\phi(x)\notin\Gamma_{H}\cdot x$.

First observe that $i(\phi(x))$, which is contained in $\Gamma_{H}$, cannot be
conjugate in $\Gamma_{H}$ to $i(x)$. Indeed, if it were, then for some
$\gamma\in\Gamma_{H}$, we would have $i(\phi(x))=\gamma
i(x)\gamma^{-1}=i(\gamma(x))$, with the second equality following from Remark
2.4. Injectivity of $i$ would then imply $\gamma(x)=\phi(x)$, a contradiction.

Hence, by item (2), there is a finite quotient $\pi:\Gamma_{H}\to F_{0}$ such
that $\pi(i(\phi(x)))$ is not conjugate in $F_{0}$ to $\pi(i(x))$. It follows
that $\pi(i(\phi(x)))\not\in\pi(i(\Gamma_{H}\cdot x))$. Indeed, by Remark 2.4,
for all $\gamma\in\Gamma_{H}$, we have that $\pi(i(\gamma(x)))$ is conjugate
in $F_{0}$ to $\pi(i(x))$, which is not conjugate to $\pi(i(\phi(x)))$.

Consider the quotient $\pi\circ i:G\to\bar{G}\leq F_{0}$. Let $K$ be the
intersection of all subgroups of $G$ with the same index as
$\mathrm{Ker}(\pi\circ i)$. This is a characteristic subgroup, and
furthermore, since $G$ is finitely generated, $K$ has finite index in $G$.
Denote the corresponding quotient map $q^{\prime\prime}:G\to G/K$. Since
$\pi\circ i$ factors through $q^{\prime\prime}$ we have
$q^{\prime\prime}(\phi(x))\notin q^{\prime\prime}(\Gamma_{H}\cdot x)$.

Since $K$ is characteristic, $q^{\prime\prime}$ induces a homomorphism
$q^{\prime}:\operatorname{Aut}(G)\to\operatorname{Aut}(G/K)$ defined by
$q^{\prime}(\gamma)(gK)=q^{\prime\prime}(\gamma(g))$. This has the property
that $q^{\prime}(\phi)\notin q^{\prime}(\Gamma_{H})$, because
$q^{\prime}(\phi)(xK)\notin q^{\prime}(\Gamma_{H})(xK)$, since the former is
$q^{\prime\prime}(\phi(x))$ and the latter is
$q^{\prime\prime}(\Gamma_{H}\cdot x)$.

The homomorphism $q^{\prime}$ descends to a homomorphism
$q:\mathrm{Out}(G)\to\mathrm{Out}(G/K)$ fitting into the following commutative
diagram:

$\begin{diagram}$

Since $q^{\prime}(\phi)\notin q^{\prime}(\Gamma_{H})$ and
$\mathrm{Inn}(G/K)=q^{\prime}(\mathrm{Inn}(G))<q^{\prime}(\Gamma_{H})$, we
have $\Psi_{K}(q^{\prime}(\phi))\notin\Psi_{K}(q^{\prime}(\Gamma_{H}))$. In
view of the commutative diagram, the former is $q(\Psi(\phi))$ and the latter
is $q(H)$, so that we found a finite quotient of $\mathrm{Out}(G)$ that
separates $\Psi(\phi)$ from $H$, namely $F=\mathrm{Out}(G/K)$, which is finite
since $K$ has finite index in $G$. ∎

##  3\. Mapping class group lemmas

In Section 3.1, we reduce condition (1) from Proposition 2.5 to a statement
about finitely many automorphisms of a surface group; this is used in the
proof of Theorem 1.1. In Section 3.2, we prove a lemma needed for verifying
malnormality of the examples from Theorem 1.2.

###  3.1. Finitely many non-pseudo-Anosovs

The goal of this subsection is to prove:

######  Proposition 3.1.

Let $\Sigma$ be a finite-type surface and let $H<MCG(\Sigma)$ be a torsion-
free malnormal convex-cocompact subgroup. Let $g\in MCG(\Sigma)-H$ have
infinite order. Then the coset $gH$ contains finitely many elements that are
not pseudo-Anosov.

The proof is postponed until after some preparatory lemmas. We will use the
following lemma to check that specific elements are pseudo-Anosov. The lemma
gives two alternative conditions, the first one is designed to show that the
product of two elements one of which acts elliptically and has "large rotation
angle" is loxodromic. The second property, instead, captures a product were
one element is loxodromic with suffieicntly large translation length.

######  Lemma 3.2 (Loxodromic products).

For every $\delta,Q\geq 0$ there exists a constant $C_{Q,\delta}$ with the
following property. Let $A$ be a group acting on a $\delta$-hyperbolic space
$X$. Let $g_{0},g_{1}\in A$ be elements that respectively stabilise $Q$-quasi-
convex sets $Y_{0},Y_{1}\subset X$. Let $\pi_{i}$ be closest-point projection
to $Y_{i}$. Assume that one of the following holds:

  1. (1)

let $y_{i}\in\pi_{i}(Y_{i+1})\subseteq Y_{i}$ and suppose that
$d(Y_{0},Y_{1})\geq C_{Q,\delta}$ and

$\operatorname{diam}_{X}(N_{3\delta}([y_{0},y_{1}])\cap
g_{i}[y_{0},y_{1}])\leq\frac{C_{Q,\delta}}{3}$

for $i=0,1$; or

  2. (2)

$diam(\pi_{1}(g_{0}Y_{1}))\leq C_{Q,\delta}/3$ and $d(g_{1}y,y)\geq
C_{Q,\delta}+2d(y,g_{0}y)$ for some $y\in Y_{1}$.

Then $g_{0}g_{1}$ is loxodromic.

###### Proof.

In case (1) we can form a bi-infinite quasi-geodesic on which $g_{0}g_{1}$
acts by translation by concatenating suitable translates of $U=[y_{1},y_{0}]$,
$V=[y_{0},g_{0}y_{0}]$, and $W=[y_{1},g_{1}y_{1}]$, with every other geodesic
being a translate of $[y_{0},y_{1}]$. Specifically, the $\langle
g_{0}g_{1}\rangle$–translates of the concatenation

$U\cdot V\cdot(g_{0}\bar{U})\cdot(g_{0}W),$

which joins $y_{1}$ to $g_{0}g_{1}y_{1}$, concatenate to form a bi-infinite
path, shown in Figure 3. Here the overline denotes the reverse path. In the
notation of Lemma 2.3, by our assumption we have $d_{i,i\pm 2}\leq
C_{Q,\delta}/3$, while the $d_{i,i\pm 1}$ are bounded independently of
$C_{Q,\delta}$ since the $[y_{0},y_{1}]$ are coarsely shortest geodesics
connecting quasiconvex sets. So Lemma 2.3 implies $g_{0}g_{1}$ is loxodromic
provided $C_{Q,\delta}$ is sufficiently large in terms of $Q$ and the constant
$C_{\delta}$ from the lemma, which depends only on $\delta$.

\begin{overpic}[width=260.17464pt]{gh} \put(12.0,5.0){$Y_{1}$}
\put(19.0,68.0){$Y_{0}$} \put(83.0,5.0){$g_{0}Y_{1}$} \put(18.0,50.0){$U$}
\put(48.0,60.0){$V$} \put(66.0,50.0){$g_{0}\bar{U}$} \put(90.0,42.0){$g_{0}W$}
\put(7.0,27.0){$y_{1}$} \put(32.0,70.0){$y_{0}$} \put(59.0,70.0){$g_{0}y_{0}$}
\put(86.0,30.0){$g_{0}y_{1}$} \put(20.0,15.0){$g_{1}^{-1}y_{1}$}
\put(101.0,41.0){$g_{0}g_{1}y_{1}$} \end{overpic} Figure 3. Proof of Lemma
3.2.(1). The paths concatenated to form the $g_{0}g_{1}$–axis are coloured
according to their $\langle g_{0}g_{1}\rangle$–orbits.

Case (2) is similar: we concatenate translates of $U=[y,g_{1}y]$ and
$V=[y,g_{0}y]$. Specifically, the $\langle g_{0}g_{1}\rangle$–translates of
the concatenation $V\cdot g_{0}U$ concatenate to form a bi-infinite path.
Again, our assumptions allow us to apply Lemma 2.3. Indeed, successive
translates of $U$ have $d_{i\pm 2}$ bounded since intersections of
neighborhoods of the relevant translates of $Y_{1}$ have diameter bounded in
terms of the diameter of $\pi_{1}(g_{0}Y_{1})$, so the assumption on
$d(y,g_{1}y)$, which controls $d_{i\pm 1}$, shows that Lemma 2.3 applies (with
the translates of $U$ as the even-indexed $\alpha_{i}$) provided
$C_{Q,\delta}$ is large enough in terms of $Q$ and $\delta$. ∎

We will also need the following lemmas to check condition 1 in Lemma 3.2 when
using it. Recall that for curve graphs we take $\delta=100$.

The following lemma can be formulated more generally for a group acting
acylindrically on a hyperbolic space and a infinite-order element whose orbits
are bounded, but we prefer to state just the special case we need to avoid
discussing coarse fixed point sets.

######  Lemma 3.3.

Let $\Sigma$ be a finite-type surface. Then there exists $D\geq 0$ with the
following property. Let $g\in MCG(\Sigma)$ be an infinite order reducible
element, fixing the clique $Y$ in $\mathcal{C}(\Sigma)$. Then for any geodesic
$\gamma$ starting in $Y$ we have that $N_{300}(\gamma)\cap g\gamma$ has
diameter at most $D$.

###### Proof.

By hyperbolicity, for every $L\geq 0$ there exists $D\geq 0$ such that the
following holds for $g,\gamma$ as in the statement: if $p\in\gamma$ is moved
distance at most $L$ by a power of $g$, then $\gamma$ has an initial
subgeodesic of length $d(p,Y)-D$ that gets moved distance at most $300$ by the
same power of $g$. Therefore, if the lemma did not hold for a sufficiently
large $D$, we would get a contradiction with acylindricity of the
$MCG(\Sigma)$–action on $\mathcal{C}(\Sigma)$. ∎

######  Lemma 3.4.

Let the group $A$ act acylindrically on the hyperbolic space $X$. Then there
exists a constant $D$ such that the following holds. Let $g\in A$ be
loxodromic and let $Y$ be its quasi-axis as in Proposition 2.1.(1). Then for
any $x\in X$ and geodesic $\gamma=[x,\pi_{Y}(x)]$ we have that
$N_{3\delta}(\gamma)\cap g\gamma$ has diameter at most $D$.

###### Proof.

This is similar to Lemma 3.3, except we "shorten" $\gamma$ on both sides. ∎

We are ready for:

###### Proof of Proposition 3.1.

Let $\Sigma,H,g$ be as in the statement and fix an $H$–orbit $H\cdot o$ in
$\mathcal{C}(S)$. Let $Q$ be the constant from Proposition 2.1. For each
nontrivial $h\in H$, let $Y_{h}$ be the $Q$–quasiconvex quasi-axis for $h$,
from Proposition 2.1.(1) (any nontrivial $h\in H$ is pseudo-Anosov). Let
$Y_{g}\subset\mathcal{C}(\Sigma)$ be the following $Q$–quasiconvex subspace:

  * •

if $g$ is reducible, then $Y_{g}$ is a clique stabilised by $g$;

  * •

if $g$ is pseudo-Anosov, then $Y_{g}$ is the quasi-axis from Proposition
2.1.(1).

Let $D$ be the maximum of the constants from Lemmas 3.3 and 3.4. Let $C_{0}$
be such that $Y_{h}\subset N_{C_{0}}(H\cdot o)$ for all $h\in H$ (which exists
since $Y_{h}$ and $H\cdot o$ are uniformly quasiconvex and $Y_{h}$ lies in
some neighborhood of $H\cdot o$). By enlarging $Q$ once, we can assume
$d(y^{\prime},gy^{\prime})\leq Q$ for all $y^{\prime}\in Y_{g}$.

We will need the following claim:

######  Claim 3.5.

For any $L\geq 0$, the set $Y_{g}\cap N_{L}(H\cdot o)$ is bounded. Moreover,
there exists $C_{1}\geq 0$ such that $\pi_{Y_{h}}(gY_{h})$ has diameter at
most $C_{1}$ for all $h\in H$.

###### Proof.

If $Y_{g}\cap N_{L}(H\cdot o)$ had unbounded diameter then $gH\cdot o$ would
have unbounded intersection with the $2L+Q$–neighborhood of $H\cdot o$, since
$Q\geq d(y^{\prime},gy^{\prime})$ for all $y^{\prime}\in Y_{g}$. Lemma 2.2
then gives that $gHg^{-1}\cap H$ is infinite, contradicting malnormality of
$H$.

Similarly, malnormality of $H$ and Lemma 2.2 imply that for all $L\geq 0$
there exists $B(L)$ such that $Y_{h}\cap N_{L}(gY_{h})$ has diameter at most
$B(L)$ (since $Y_{h}\subset N_{C_{0}}(H\cdot o)$ and $gY_{h}\subset
N_{C_{0}}(gH\cdot o)$ and the lemma bounds the diameter of coarse
intersections of the two coset orbits). It is well-known that quasiconvex sets
in a hyperbolic space that have bounded intersection of neighborhoods also
have bounded projection onto each other; we sketch this argument in our
notation.

Consider two points $a,b\in gY_{h}$ and
$a^{\prime}\in\pi_{Y_{h}}(a),b^{\prime}\in\pi_{Y_{h}}(b)$. Any point on
$[a^{\prime},b^{\prime}]$ which is further away than $500+2Q$ from
$\\{a^{\prime},b^{\prime}\\}$ cannot be $200$-close to $[a,a^{\prime}]$ or
$[b,b^{\prime}]$, otherwise $a^{\prime}$ would not be nearly closest to $a$
(that is, $a^{\prime}\notin\pi_{Y_{h}}(a)$) or $b^{\prime}$ would not be
nearly closest to $b$. Hence, any such point is within $200$ of $[a,b]$ by
thinness of quadrilaterals. Hence such point is within $200+Q$ from $gY_{h}$,
as well as within $Q$ of $Y_{h}$. Hence, the diameter of $Y_{h}\cap
N_{L}(gY_{h})$ for $L=200+2Q$ is at least
$d(a^{\prime},b^{\prime})-10^{3}-10Q$. Since $a^{\prime},b^{\prime}$ were
arbitrary, the diameter of $\pi_{Y_{h}}(gY_{h})$ is bounded in terms of
$B(L)$. ∎

Finally, let $C_{Q,100}$ be the constant from Lemma 3.2. We can assume that
$C_{Q,100}\geq 10D+10^{3}+3C_{1}$.

Fix $h\in H$. First suppose that $d(Y_{g},Y_{h})>C_{Q,100}$. Then Lemma
3.2.(1) applies in view of Lemma 3.4 (if $g$ is pseudo-Anosov), or Lemma 3.3
(if $g$ is infinite-order reducible). Hence Lemma 3.2.(1) shows that $gh$ is
pseudo-Anosov.

Next, suppose that the translation length of $h$ satisfies
$\tau_{h}>2Q+10C_{Q,100}$ and that $d(Y_{g},Y_{h})\leq C_{Q,100}$. Recall that
$diam(\pi_{Y_{h}}(gY_{h}))\leq C_{1}\leq C_{Q,100}/3$. Choose $y\in Y_{h}$
within distance $C_{Q,100}$ of some $y^{\prime}\in Y_{g}$. Hence

$2d(y,gy)+C_{Q,100}\leq 2d(y^{\prime},gy^{\prime})+5C_{Q,100}<\tau_{h}\leq
d(y,hy).$

So we can apply Lemma 3.2.(2) (with $g_{1}=h$ and $g_{0}=g$) and see that $gh$
is pseudo-Anosov.

To conclude, it suffices to prove the following:

######  Claim 3.6.

There exist finitely many $h$ such that $d(Y_{g},Y_{h})\leq C_{Q,100}$ and
$\tau_{h}\leq 2Q+10C_{Q,100}$.

###### Proof.

By Claim 3.5 (which says that $Y_{g}$ can only come close to $H\cdot o$ at a
bounded set) and since $Y_{h}$ lies within $C_{0}$ of $H\cdot o$, we can only
have $d(Y_{g},Y_{h})\leq C_{Q,100}$ if $d(Y_{h},o)\leq C^{\prime}$ for some
$C^{\prime}$ (depending on $g$ also).

Since $H$ is quasi-isometrically embedded in $\mathcal{C}(\Sigma)$, it
suffices to show that in (a fixed Cayley graph of) a hyperbolic group $G$, for
any $B\geq 0$ there are only finitely many $h\in G$ with translation distance
bounded by $B$ and such that $Y_{h}$ contains a point within $B$ of the
identity (note that quasi-axes in $G$ get mapped within bounded distance of
quasi-axes in $\mathcal{C}(\Sigma)$, and translation lengths get distorted in
a way controlled by the quasi-isometry constants). This is true since any such
$h$ moves the identity a uniformly bounded amount. ∎

We have shown that for all $h\in H$, either $gh$ is pseudo-Anosov or $h$ is
one of the finitely many elements of $H$ from Claim 3.6, which proves the
proposition. ∎

###  3.2. Pseudo-Anosovs without hidden symmetries

######  Lemma 3.7.

Let $\Sigma$ be a finite-type hyperbolic surface with genus $\mathfrak{g}$ and
$\mathfrak{p}$ punctures, with
$(\mathfrak{g},\mathfrak{p})\notin\\{(1,1),(1,2),(2,0)\\}$. Then there exists
a pseudo-Anosov $\phi\in MCG(\Sigma)$ and $n_{0}\geq 0$ such that the
following holds. For any curve $c$ on $\Sigma$ and $n\geq n_{0}$ we have
$\operatorname{Stab}(c)\cap\operatorname{Stab}(\phi^{n}(c))=\\{1\\}$.

###### Proof.

Proposition 2.1.(2) yields, for a pseudo-Anosov $g$, a unique maximal
elementary subgroup $E(g)$ containing $g$ and an invariant quasi-geodesic
$\gamma_{g}$ in the curve graph $\mathcal{C}(\Sigma)$, with the following
property. For every $D\geq 0$ there exists $L\geq 0$ such that if $h\in G$ has
the property that there exist $x,y\in\gamma_{g}$ with
$d_{\mathcal{C}(\Sigma)}(x,y)\geq L$ and
$d_{\mathcal{C}(\Sigma)}(x,h(x)),d_{\mathcal{C}(\Sigma)}(y,h(y))\leq C$, then
$h\in E(g)$.

Let $\phi$ be a pseudo-Anosov for which $E(\phi)=\langle\phi\rangle$. Such a
$\phi$ exists by [DGO17, Theorem 6.14], since the maximal finite normal
subgroup of a mapping class group coincides with its centre (see the proof of
[MT21, Proposition 11.15]) and the centre is trivial under our assumptions on
$(\mathfrak{g},\mathfrak{p})$ (see [FM12, Section 3.4]).

Given any curve $c$ and any sufficiently large $n$, if
$h\in\operatorname{Stab}(c)\cap\operatorname{Stab}(\phi^{n}(c))$, then $h$
coarsely stabilises the closest-point projections of $c$ and $\phi^{n}(c)$ to
$\gamma_{\phi}$, so that $h\in E(\phi)=\langle\phi\rangle$ by the above. No
non-trivial power of $\phi$ fixes a curve, so $h=1$. ∎

##  4\. Common witnesses for non-inner automorphisms

######  Proposition 4.1.

Let $G$ be the fundamental group of a closed connected orientable surface of
finite genus, and let $\phi_{1},\dots,\phi_{n}$ be non-inner automorphisms of
$G$. Then there exists $x\in G$ such that for all $i$, the elements
$\phi_{i}(x)$ and $x$ are not conjugate.

We need the following version of [MO10, Lemma 4.4] (which we use in the
proof).

######  Lemma 4.2.

Let $G$ be a torsion-free hyperbolic group. Let $l\geq 2$ and let
$\\{g_{1},\dots,g_{l}\\}$, $\\{h_{1},\dots,h_{l}\\}$ be sets of non-
commensurable, primitive, infinite order elements. Assume that if
$g_{1}^{m_{1}}\dots g_{l}^{m_{l}}$ is the identity of $G$ for some $m_{i}$
then all $m_{i}$ are $0$.

There exists $N\in\mathbb{N}$ such that the following holds. Suppose that

$g=g_{1}^{n_{1}}g_{2}^{n_{2}}\dots g_{l}^{n_{l}}$

is conjugate to

$h=h_{1}^{n_{1}}\cdots h_{l}^{n_{l}}$

in $G$, and for all $1\leq i\leq l$ we have $n_{i}\geq N$. Then there exists
$x\in G$ and $k\in\mathbb{N}$ with $h_{i}=xg_{i+k}x^{-1}$ for all $i$ (indices
mod $l$).

###### Proof.

We conflate $G$ with a fixed Cayley graph.

There exists a constant $C_{0}$ such that for all sufficiently large $N$ we
have that $g$ stabilises a $(C_{0},C_{0})$–quasigeodesic $\alpha_{g}$ obtained
by concatenating translates of the discrete paths
$\\{1,g_{i},g_{i}^{2},\dots,g_{i}^{n_{i}}\\}$, with $n_{i}\geq N$, that we
call _pieces_ , and similarly for $h$. This is because the $\langle
g_{i}\rangle$ are quasi-geodesic lines, and moreover any two translates of two
distinct such quasi-geodesic lines cannot stay within bounded distance of each
other for arbitrarily long time in view of Proposition 2.1.(2) ($g_{i}$ being
not commensurable to $g_{j}$ is equivalent to $g_{i}\notin E(g_{j})$).

Let $D\geq 0$ be such that any two $(C_{0},C_{0})$–quasigeodesics in $G$ with
the same endpoints at infinity lie within distance $D$ of each other.

If $g$ and $h$ are conjugate, we have that two translates of $\alpha_{g}$ and
$\alpha_{h}$ have the same endpoints at infinity. Hence, for all $M$ the
following holds for all large $N$: each piece for $h$ has a subpath of length
at least $M$ that has a translate that lies within $D$ of a piece for
$\alpha_{g}$. See Figure 4.

\begin{overpic}[width=433.62pt]{pieces} \put(62.0,6.0){$D$}
\put(36.0,6.0){$D$} \put(49.0,-2.0){$M$} \end{overpic} Figure 4. Edges in the
bottom axis $\alpha_{h}$ are labelled by the $h_{i}$, with a translate of the
piece $\\{1,h_{i},\ldots,h_{i}^{5}\\}$ shown in the middle. This has a long
subpath $D$–close to a subpath of the piece
$\\{1,g_{\sigma(i)},\ldots,g_{\sigma(i)}^{5}\\}$ of the translate of
$\alpha_{g}$ shown at the top. (Large dots represent concatenation points of
successive pieces.)

For $M$ large enough, this implies that each $h_{i}\in
x_{i}E(g_{\sigma(i)})x^{-1}_{i}$ for some $\sigma(i)\in\\{1,\ldots,l\\}$ and
some $x_{i}\in G$ by Proposition 2.1.(2). Note that $i\mapsto\sigma(i)$ is
injective since $\\{h_{1},\ldots,h_{l}\\}$ are pairwise non-commensurable.

To sum up, either there is no bijection $\sigma$ and elements $x_{i}$ such
that $h_{i}$ lies in $x_{i}E(g_{\sigma(i)})x^{-1}_{i}$, in which case $g$ and
$h$ cannot be conjugate for any large $N$ and the conclusion holds vacuously,
or there exist $\sigma$ and $x_{i}$, in which case we can fix them and they
only depend on the choice of $g_{i}$ and $h_{i}$.

We now fix $\sigma$ and $x_{i}$ as above. We further let $F$ be the (finite)
collection of all elements of the form $x_{i}^{-1}x_{i+1}$ (indices mod $l$).
Therefore, $h$ (whence $g$) is conjugate to
$(h^{\prime}_{1})^{n_{1}}f_{1}\cdots(h^{\prime}_{l})^{n_{l}}f_{l}$, for some
$h^{\prime}_{i}\in E(g_{\sigma(i)})$, where $f_{i}=x_{i}^{-1}x_{i+1}$ (indices
mod $l$). We have now reduced to the setup of [MO10, Lemma 4.4], from which we
can conclude that $\sigma$ is a cyclic shift, say $\sigma(i)=i+k$ for some
fixed $k$, and $f_{i}\in E(g_{\sigma(i)})E(g_{\sigma(i+1)})$.

Now, we have $E(g_{j})=\langle g_{j}\rangle$ for all $j$ by the primitivity
hypothesis and the assumption that $G$ is torsion-free, so
$f_{i}=g_{\sigma(i)}^{l_{i}}g_{\sigma(i+1)}^{m_{i}}$ for some $l_{i},m_{i}$.
Up to multiplying each $x_{i}$ on the right by $g_{\sigma(i)}^{l_{i}}$ (which
does not affect the equation $h_{i}=x_{i}h^{\prime}_{i}x_{i}^{-1}$), we then
get $x_{i+1}=x_{i}g_{\sigma(i+1)}^{m^{\prime}_{i}}$, where
$m^{\prime}_{i}=m_{i}+l_{i+1}$. This holds with indices mod $l$, so that
$x_{1}=x_{1}g_{\sigma(2)}^{m^{\prime}_{1}}\dots
g_{\sigma(l)}^{m^{\prime}_{l-1}}g_{\sigma(1)}^{m^{\prime}_{l}}$. By our
hypothesis on the $g_{i}$ (applied to a cyclic conjugate) we get that all
$m_{i}$ are $0$, and therefore we get that all $x_{i}$ are equal, giving us
the required $x$. ∎

With the lemma in hand, we proceed to:

###### Proof of Proposition 4.1.

Fix some $\phi=\phi_{i}$, and let $g_{1},\dots,g_{l}$ be the standard
generators of the surface group $G$. Observe that the condition from Lemma 4.2
about products of the form $g_{1}^{m_{1}}\dots g_{l}^{m_{l}}$ holds in this
setup. Indeed, the $g_{i}$ are generators in a $C^{\prime}(1/6)$ small
cancellation presentation where the (only) relator $R$, as a cyclic word, has
the following property. Any subword of a cyclic permutation of $R$ which has
length more than half the length of $R$ is not a subword of a cyclic
permutation of a product $g_{1}^{m_{1}}\dots g_{l}^{m_{l}}$. We can conclude
in view of Greendlinger’s lemma.

Let $h_{j}=\phi(g_{j})$. Consider the elements $g=g_{1}^{T}g_{2}^{T+1}\dots
g_{l}^{T+l-1}$ and $h=h_{1}^{T}h_{2}^{T+1}\dots h_{l}^{T+l-1}$, for $T$ larger
than the $N$ from Lemma 4.2. We claim that $g$ and $h$ cannot be conjugate,
which implies that we can take $x=g$ for some sufficiently large $T$ which
works simultaneously for all $\phi_{i}$.

If $g$ and $h$ were conjugate, then we would have $h_{i}=xg_{i+k}x^{-1}$ for
some fixed $x\in G$ and natural number $k<l$. We claim that $k=0$. This holds
since there is a homomorphism $\psi$ to $\mathbb{Z}$ such that $\psi(g_{1})=1$
and $\psi(g_{j})=0$ for all $j\neq 1$, so that $\psi(g)=T$ and $\psi(h)=T+k$,
so $k=0$. Hence $\phi(g_{i})$ acts as conjugation by $x$ on all generators,
implying that it is inner, contradicting the hypotheses. ∎

Now we can prove the separability criterion.

###### Proof of Theorem 1.1.

Throughout the proof we think of the mapping class group as an index 2
subgroup of $\mathrm{Out}(\pi_{1}(\Sigma_{\mathfrak{g}}))$. Let
$\Psi:\operatorname{Aut}(\pi_{1}(\Sigma_{\mathfrak{g}}))\to\mathrm{Out}(\pi_{1}(\Sigma_{\mathfrak{g}}))$
be as in Section 2.5.

Since $MCG(\Sigma_{\mathfrak{g}})$ is residually finite [Gro75] and has
finitely many conjugacy classes of finite-order elements [Bri00], there is a
finite-index torsion-free subgroup $M_{0}\leq MCG(\Sigma_{\mathfrak{g}})$.

######  Claim 4.3.

Suppose that $\phi\in\operatorname{Aut}(\pi_{1}(\Sigma_{\mathfrak{g}}))$
satisfies $\bar{\phi}=\Psi(\phi)\in M_{0}-(H\cap M_{0})$. Then there is a
finite group $F$ and a quotient
$q:\mathrm{Out}(\pi_{1}(\Sigma_{\mathfrak{g}}))\to F$ such that
$q(\bar{\phi})\notin q(H)$.

###### Proof of Claim 4.3.

We will use Proposition 2.5. First, by hypothesis, $\Gamma_{H}=\Psi^{-1}(H)$
is conjugacy separable, so condition (2) of Proposition 2.5 is satisfied. So,
we are left to verify condition (1) of Proposition 2.5 for the given $\phi$.

Since $\bar{\phi}\in M_{0}-(H\cap M_{0})$, the element $\bar{\phi}$ has
infinite order in $MCG(\Sigma_{\mathfrak{g}})$, so Proposition 3.1 applies.
The only properties of $\phi$ used in the following argument are that
$\bar{\phi}\in MCG(\Sigma)$, and $\bar{\phi}$ has infinite order, and
$\bar{\phi}\not\in H$.

We have to find $x\in\pi_{1}(\Sigma_{\mathfrak{g}})$ such that $\phi(x)\neq
h(x)$ for $h\in\Gamma_{H}$. This is equivalent to $\phi^{-1}h(x)\neq x$.

If $\Psi(\phi^{-1}h)$ (which is necessarily in $MCG(\Sigma_{\mathfrak{g}})$
since $\Psi(h)\in H$ and $\bar{\phi}\in MCG(\Sigma_{\mathfrak{g}})$) is
pseudo-Anosov, then any nontrivial $x$ satisfies the required condition for
the $h$ in question. Hence, we are only concerned with the non-pseudo-Anosov
elements of $\bar{\phi}^{-1}H$, of which there are only finitely many by
Proposition 3.1. Taking arbitrary preimages of these elements, we obtain
finitely many $h_{i}\in\phi^{-1}\Gamma_{H}$ such that if $x\in G$ has the
property that $h_{i}(x)$ is not conjugate to $x$ for all $i$, then $x$
satisfies the required property, that is, $\phi(x)\neq h(x)$ for all
$h\in\Gamma_{H}$.

The existence of such $x$ is guaranteed by Proposition 4.1. Hence Proposition
2.5 supplies a finite quotient
$q:\mathrm{Out}(\pi_{1}(\Sigma_{\mathfrak{g}}))\to F$ such that
$q(\bar{\phi})\not\in q(H)$. ∎

From Claim 4.3, we get that $H\cap M_{0}$ is separable in $M_{0}$, since for
each $\phi\in M_{0}-H\cap M_{0}$, we can restrict the quotient $q$ from the
claim to $M_{0}\leq\mathrm{Out}(\pi_{1}(\Sigma_{\mathfrak{g}}))$ to get a
finite quotient $q:M_{0}\to\bar{M}_{0}\leq F$ where $q(\phi)\not\in q(H\cap
M_{0})$. In summary, we have a finite-index subgroup $M_{0}\leq
MCG(\Sigma_{\mathfrak{g}})$ and a subgroup $H\leq MCG(\Sigma_{\mathfrak{g}})$
such that $H\cap M_{0}$ is separable in $M_{0}$. Applying Proposition 2.2.(ii)
of [Nib92] (taking $K=K_{0}=\\{1\\}$ in the statement of that proposition)
shows that $H$ is separable in $MCG(\Sigma_{\mathfrak{g}})$, as required. ∎

##  5\. MMS subgroups

We finally construct examples to which Theorem 1.1 applies. We will use the
following form of the Bounded Geodesic Image Theorem. We denote the subsurface
projection to the subsurface $Y$ by $\pi_{Y}$ and abbreviate
$d_{\mathcal{C}(Y)}(\pi_{Y}(\cdot),\pi_{Y}(\cdot))=d_{\mathcal{C}(Y)}(\cdot,\cdot)$.

######  Theorem 5.1.

[MM00, Theorem 3.1] For each $\mathfrak{g}\geq 2$ there exists $B\geq 0$ such
that the following holds. If the vertices $x,y\in\mathcal{C}(\Sigma)$ satisfy
$d_{\mathcal{C}(\Sigma_{\mathfrak{g}}-v)}(x,y)\geq B$ for some non-separating
curve $v$, then any geodesic from $x$ to $y$ contains $v$.

We now prove Theorem 1.2:

######  Theorem 5.2.

For every $\mathfrak{g}\geq 3$ and $n\geq 1$ there exist infinitely many
conjugacy classes of malnormal convex cocompact subgroups $H$ of
$MCG(\Sigma_{\mathfrak{g}})$ isomorphic to a free group of rank $n$ whose
corresponding extension $\Gamma_{H}$ is virtually compact special.

###### Proof.

In [MMS19] the authors construct subgroups (that we call _MMS subgroups_)
satisfying all requirements except possibly malnormality. MMS subgroups are
described in [MMS19, Proposition 5.7, Theorem 5.8]. Our goal is to construct
(many) malnormal MMS subgroups.

Each MMS subgroup $H$ stabilises a _tight tree of homologous non-separating
curves_ (a _tight tree_ for short), which is a tree $T$ with the following
description (see [MMS19, Section 3.1]):

  * •

For each vertex $v$ of $T$ we have a vertex
$c_{v}\in\mathcal{C}(\Sigma_{\mathfrak{g}})$, and for each edge $e$ of $T$, we
have a tight geodesic $\gamma_{e}$ in $\mathcal{C}(\Sigma_{\mathfrak{g}})$
connecting the vertices corresponding to the endpoints of $e$.

  * •

All multicurves occurring along $\gamma_{e}$ (including the endpoints) are
formed from non-separating curves. Moreover, all curves appearing along all
tight geodesics are homologous to each other.

There are some additional properties of $T$ guaranteeing that the natural
subdivision of $T$ isometrically embeds into
$\mathcal{C}(\Sigma_{\mathfrak{g}})$ ([MMS19, Proposition 3.2]). Moreover, the
tight tree $T$ stabilised by an MMS subgroup $H$ has the property that
whenever $u,v,w$ are consecutive vertices along a tight geodesic as above,
then there is a pseudo-Anosov on $\Sigma_{\mathfrak{g}}-\\{v\\}$ which maps
$u$ to $w$. In fact, the tree is constructed starting from a choice of such
partial pseudo-Anosovs and any choice of sufficiently large powers of the
partial pseudo-Anosovs ([MMS19, Section 5.1]).

As a result, there are MMS subgroups $H$ that stabilise a tight tree $T$ with
the following additional properties. For an edge $e$ of the tree, we call the
_projection numbers_ the quantities
$d_{\mathcal{C}(\Sigma_{\mathfrak{g}}-v)}(u,w)$ for consecutive $u,v,w$ along
$\gamma_{e}$. Then

  * •

for any edge $e$ of $T$, all projection numbers are different and larger than
$10B$, for $B$ as in Theorem 5.1,

  * •

for any two edges $e,f$ of $T$ belonging to different $H$-orbits, all
projection numbers for $e$ are different from all projection numbers for $f$,

  * •

the tight geodesics all have length at least $10^{3}$,

  * •

$Stab_{MCG(\Sigma_{\mathfrak{g}}-v)}(u)\cap
Stab_{MCG(\Sigma_{\mathfrak{g}}-v)}(w)$ is trivial for all $u,v,w$ that are
consecutive in $\gamma_{e}$, where $e$ is any edge of $T$.

The last item can be arranged using the pseudo-Anosovs constructed in Lemma
3.7.

We now verify malnormality of $H$. Suppose for some $g\in
MCG(\Sigma_{\mathfrak{g}})$ that $gHg^{-1}\cap H$ is infinite. Then $T$ and
$gT$ respectively contain bi-infinite geodesics $\alpha$ and
$g\alpha^{\prime}$ that 200-fellow-travel.

Fix an edge $e$ of $T$ such that $\gamma_{e}\subset\alpha$. We claim that for
all vertices $c\in\gamma_{e}$ that are at distance more than $200$ from both
endpoints of $\gamma_{e}$, the geodesic $g\alpha^{\prime}$ passes through $c$.
Notice that then the third item provides at least 100 such vertices $c$.

To prove the claim we use the lower bound on projection numbers from the first
of the above four items and Theorem 5.1, as follows. First of all, by Theorem
5.1 the endpoints $a^{\prime},b^{\prime}$ of $\gamma_{e}$ each project to
$\mathcal{C}(\Sigma_{\mathfrak{g}}-c)$ within $B$ of the projection of one of
the vertices $a,b$ adjacent to $c$ along $\gamma_{e}$. Again by Theorem 5.1
points $a^{\prime\prime},b^{\prime\prime}$ in $g\alpha^{\prime}$ that lie
within $200$ of $a^{\prime},b^{\prime}$ project within $2B$ of the projections
of $a,b$. In particular, $a^{\prime\prime},b^{\prime\prime}$ project $B$-far
on $C(\Sigma_{\mathfrak{g}}-c)$, and once again by Theorem 5.1 we have that
$g\alpha^{\prime}$ passes through $c$.

With another application of the third item, there exists an edge $f$ of $T$
such that there are distinct triples $u,v,w$ and $u_{1},v_{1},w_{1}$ of
$\gamma_{e}$–consecutive vertices, both of which are contained in
$g\gamma_{f}$.

By the second item, $e$ and $f$ are in the same $H$–orbit. Hence there exists
$h\in H$ such that $u,v,w,u_{1},v_{1},w_{1}\in\gamma_{e}\cap(gh)\gamma_{e}$.

By the "projection numbers are different" clause of the first item,
$ghu=u,ghv=v,ghw=w$. From the fourth item, it follows that $gh$ is a power of
the Dehn twist around $v$.

The same argument applied to $u_{1},v_{1},w_{1}$ shows that $gh$ is also a
power of the Dehn twist around $v_{1}$, so $gh=1$ and thus $g\in H$, i.e. $H$
is malnormal.

To arrange infinitely many conjugacy classes the argument is similar,
arranging the projection numbers to be different for tight trees associated to
different subgroups. ∎

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