Quotients of mapping class groups from Out(Fn )

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arXiv:1608.03868v3 [] 27 Mar 2018

Quotients of mapping class groups from Out(Fn)
Khalid Bou-Rabee∗

Christopher J. Leininger †

March 29, 2018

Abstract
We give a short proof of Masbaum and Reid’s result that mapping class groups involve any
finite group, appealing to free quotients of surface groups and a result of Gilman, following
Dunfield–Thurston.
keywords: mapping class groups, involve, finite groups

Let Σg be a closed oriented surface of genus g and Fn a nonabelian free group of rank n. The
fundamental group, π1 (Σg ), is residually free [Bau62] and Fn has a wealth finite index subgroups
[MKS04, pp. 116]. In [DT06], N. Dunfield and W. Thurston consider the action of the mapping class
group Mod(Σg ) on the set of finite index normal subgroups of π1 (Σg ) with finite simple quotients,
and in particular those containing the kernel of an epimorphism π1 (Σg ) → Fg . Their observations
relating to work of R. Gilman [Gil77], give rise to finite index subgroups of Mod(Σg ) that surjects
symmetric groups of arbitrarily large order; see the discussion following the proof of [Bau62, Theorem 7.4].
Theorem 1 (Dunfield–Thurston). For all g ≥ 3, r ≥ 1, there exists an epimorphism φ : π1 (Σg ) → Fg
and a prime q, so that

N ⊳ π1 (Σg ) | ker φ < N and π1 (Σg )/N ∼
= PSL(2, q)

has at least r elements, and its (finite index) stabilizer in Mod(Σg ) acts as the full symmetric group
on this set.
We explain the proof of this in Section 1.2. In this note, we observe that since every finite group
embeds in some finite symmetric group, Theorem 1 provides a new elementary proof of a result of
G. Masbaum and A. Reid [MR12]. Recall that a group G involves a group H if there exists a finite
index subgroup L ≤ G and a surjective map φ : L → H.
Corollary 2 (Masbaum–Reid). Let Σg,m be a surface of genus g with m punctures. If 3g − 3 + m ≥ 1
(or g = 1 and m = 0) then Mod(Σg,m ) involves any finite group.
The few mapping class groups not covered by the corollary are finite groups; see, e.g. [FM12].
Corollary 2 is also proved using arithmetic methods by F. Grunewald, M. Larsen, A. Lubotzky, and
J. Malestein [GLLM15].
Further applications of the quotients from Theorem 1 include new proofs of residual finiteness
and separability of handlebody groups; see §3 for theorem statements and proofs.
Acknowledgements. The authors would like to thank Alan Reid and Alex Lubotzky for helpful
conversations and Benson Farb for suggesting improvements on an earlier draft.
∗ K.B.
† C.L.

supported in part by NSF grant DMS-1405609
supported in part by NSF grant DMS-1510034.

Quotients of mapping class groups from Out(Fn )

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1 Preliminaries
1.1 G-defining subgroups
Here we collect some results surrounding definitions and discussions in R. Gilman [Gil77]. Let
G and F be groups. A G-defining subgroup of F is a normal subgroup N of F such that F/N is
isomorphic to G. Let X(F, G) denote the set of all G–defining subgroups of F. The automorphism
group Aut(F) acts on normal subgroups of F while preserving their quotients, and hence on the set
X(F, G) of G-defining subgroups of F. Since inner automorphisms act trivially, the action descends
to an action of the outer automorphism group of F, Out(F), on X(F, G). If G is finite, and F is
finitely generated, one obtains a finite permutation representation of Out(F). Let Fn be the free
group of rank n. The following is Theorem 1 of [Gil77].
Theorem 3 (Gilman). For any n ≥ 3 and prime p ≥ 5, Out(Fn ) acts on the PSL(2, p)-defining
subgroups of Fn as the alternating or symmetric group, and both cases occur for infinitely many
primes.
From the proof, Gilman obtains the following strengthened form of residual finiteness for Out(Fn ).
Corollary 4 (Gilman). For any n ≥ 3, the group Out(Fn ) is residually finite alternating and residually finite symmetric via the quotients from Theorem 3.
This means that for any φ ∈ Out(Fn ) − {1}, there exist primes p so that the action of Out(Fn )
on X(Fn , PSL(2, p)) is alternating (and also primes p so that the action is symmetric), and φ acts
nontrivially.
We will also need the following well-known fact, obtained from the classical embedding of a
free group into PSL(2, Z) as a subgroup of finite index, (c.f. A. Peluso [Pel66]).
Lemma 5. For any n ≥ 2, any element α ∈ Fn − {1}, and all but finitely many primes p, there exists
a PSL(2, p)–defining subgroup of Fn not containing α .
Proof. Let Fn be a finite index, free subgroup of rank n in the free group F2 := ha, bi. Identify Fn
with its image in PSL(2, Z) under the injective homomorphism F2 → PSL(2, Z) given by




1 2
1 0
a 7→
, and b 7→
.
0 1
2 1
Let α ∈ Fn − {1} be given and let A ∈ SL(2, Z) be a matrix representing α . Since α 6= 1, we may
assume that either A has a nonzero off-diagonal entry d 6= 0, or else a diagonal entry d > 1. Then for
any prime p not dividing d in the former case, or d ± 1 in the latter, we have that π p (α ) is nontrivial
in the quotient π p : PSL(2, Z) → PSL(2, p); that is, α ∈
/ ker π p .
Since Fn has finite index in F2 , there exists m ≥ 1 so that the matrices




1 m
1 0
and
0 1
m 1
represent elements of Fn in PSL(2, Z). For any prime p not dividing m, the π p –image of these
elements generate PSL(2, p). Thus, for all but finitely many primes p, ker π p ∩ Fn is a PSL(2, p)–
defining subgroup not containing α .

1.2 Handlebody subgroups and maps to free groups
Let Σ = Σg be a closed surface of genus g ≥ 2 and let H = Hg be a handlebody of genus g. Given a
homeomorphism φ : Σ → ∂ H ⊂ H, the induced homomorphism is a surjection φ∗ : π1 (Σ) → π1 (H) ∼
=
Fg . As is well-known, every epimorphism π1 (Σ) → Fg arises in this way (see e.g. Lemma 2.2 in
[LR02]). The kernel, ∆φ = ker(φ∗ ) is an Fg –defining subgroup, and is the subgroup generated by the

Quotients of mapping class groups from Out(Fn )

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simple closed curves on Σ whose φ –images bound disks in H. We write Hφ for the handlebody H,
equipped with the homeomorphism φ : Σ → ∂ H.
Let Mod(Hφ ) denote the subgroup of the mapping class group Mod(Σ) consisting of the isotopy
classes of homeomorphisms that extend over Hφ (via the identification φ : Σ → ∂ H). Equivalently,
Mod(Hφ ) consists of those mapping classes [ f ] such that f∗ (∆φ ) = ∆φ ; that is Mod(Hφ ) is the
stabilizer in Mod(Σ) of ∆φ . Any element [ f ] ∈ Mod(Hφ ) induces an automorphism we denote
Φ∗ ([ f ]) ∈ Out(Fg ), which defines a homomorphism Φ∗ : Mod(Hφ ) → Out(Fg ). The main result of
[Gri64] implies the next proposition.
Proposition 6. For any g ≥ 0, and homeomorphism φ : Σ → ∂ H, Φ∗ : Mod(Hφ ) → Out(Fg ) is
surjective.
The kernel of Φ∗ , the set of mapping classes in Mod(Hφ ) that act trivially on π1 (H) is also a
well-studied subgroup denoted Mod0 (Hφ ).
Recall that X(Fg , G) and X(π1 (Σ), G) are the sets of G–defining subgroups of Fg and π1 (Σ),
respectively. Define
X φ (π1 (Σ), G) := {φ∗−1 (N) | N ∈ X(Fg , G)} ⊂ X(π1 (Σ), G),
Alternatively, this is precisely the set of G–defining subgroups containing ∆φ :
X φ (π1 (Σ), G) = {N ∈ X(π1 (Σ), G) | ∆φ < N}.
Lemma 7. The handlebody subgroup is contained in the stabilizer
Mod(Hφ ) < stab X φ (π1 (Σ), G).
Moreover, if Out(Fg ) acts on X(Fg , G) as the full symmetric group, then Mod(Hφ ) (and hence
stab X φ (π1 (Σ), G)) acts on X φ (π1 (Σ), G) as the full symmetric group.
Proof. Let N ∈ X φ (π1 (Σ), G) and let [ f ] ∈ Mod(Hφ ), where f is a representative homeomorphism.
Since f∗ (∆φ ) = ∆φ , we have ∆φ < f∗ (N), and f∗ (N) ∈ X φ (π1 (Σ), G). Thus, f∗ preserves X φ (π1 (Σ), G),
as required.
The last statement follows immediately from Proposition 6 and the fact that the bijection from
the correspondence theorem X φ (π1 (Σ), G) → X(Fg , G) is Φ∗ –equivariant.

2 Mapping class groups involve any finite group: The proofs of
Theorem 1 and Corollary 2.
Here we give the proof of Theorem 1, following Dunfield–Thurston (see [DT06, pp. 505-506]).
Proof of Theorem 1. Fix g ≥ 3 and let Π be the infinitely many primes for which Out(Fg ) acts on the
PSL(2, p)-defining subgroups as the symmetric group, guaranteed by Theorem 3. As a consequence
of Corollary 4, the cardinality of X(Fg , PSL(2, p)) is unbounded over any infinite set of primes p,
and hence there exists a prime p ∈ Π where the number of PSL(2, p)-defining subgroups is R ≥ r.
By Lemma 7, stab X φ (π1 (Σ), PSL(2, p)) acts on X φ (π1 (Σ), PSL(2, p)) as the symmetric group,
defining a surjective homomorphism
stab X φ (π1 (Σ), PSL(2, p)) → Sym(X φ (π1 (Σ), PSL(2, p))) ∼
= SR .
Since X(π1 (Σ), PSL(2, p)) is a finite set, stab X φ (π1 (Σ), PSL(2, p)) < Mod(Σ) has finite index, completing the proof.

Quotients of mapping class groups from Out(Fn )

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Proof of Corollary 2. For g, m as in the statement and any r ∈ N, we show that there is a finite index
subgroup of Mod(Σg,m ) that surjects a symmetric group on at least r elements. Since any finite
subgroup is isomorphic to a subgroup of some such symmetric group, this will prove the theorem.
First observe that for any m, g ≥ 0, the kernel of the action of Mod(Σg,m ) on the m punctures
of Σg,m is a finite index subgroup Mod′ (Σg,m ) < Mod(Σg,m ). Furthermore, if 0 ≤ m < m′ , there
is a surjective homomorphism Mod′ (Σg,m′ ) → Mod′ (Σg,m ) obtained by “filling in” m′ − m of the
punctures; see [FM12].
Now, because Mod′ (Σ0,4 ) ∼
= F2 , and the symmetric group on r elements is 2–generated, it follows
that Mod′ (Σ0,4 ) surjects Sr . From the previous paragraph, it follows that Mod′ (Σ0,m ) surjects Sr for
all m ≥ 4. Similarly, Mod(Σ1,0 ) ∼
= SL(2, Z), which has a finite index subgroup isomorphic to F2 ,
and so there is a finite index subgroup of Mod(Σ1,m ) that surjects Sr for all m ≥ 0. As shown
in [BH71], there is a surjective homomorphism Mod(Σ2,0 ) → Mod(Σ0,6 ), and consequently, we
may find surjective homomorphisms from finite index subgroups of Mod(Σ2,m ) to Sr for all m ≥
0. From this and the previous paragraph, it suffices to assume g ≥ 3 and m = 0. The required
surjective homomorphism to a symmetric group in this case follows from Theorem 1, completing
the proof.

3 Separating handlebody subgroups and residual finiteness
The finite quotients of Mod(Σ) coming from surjective homomorphisms π1 (Σg ) → Fg also allow us
to deduce the following result of [LM07]. Recall that a subgroup K < F is said to be separable in F
if for any α ∈ F − K, there exists a finite index subgroup G < F containing K and not containing α .
Theorem 8 (Leininger-McReynolds). For any g ≥ 2 and homeomorphism to the boundary of a
handlebody, φ : Σ → ∂ H, the groups Mod(Hφ ) and Mod0 (Hφ ) are separable in Mod(Σg ).
While the proof of separability of Mod(Hφ ) in Mod(Σg ) works for all g ≥ 2, separability of Mod0 (Σg )
only follows from the discussion here when g ≥ 3.
Proof. Suppose Σ = Σg for g ≥ 2, and observe that for any p, any [h] ∈ stab X φ (π1 (Σ), PSL(2, p)),
and any α ∈ ∆φ , we have h∗ (α ) ∈ K for all K ∈ X φ (π1 (Σ), PSL(2, p)). By Lemma 7 this is true for
all [h] ∈ Mod(Hφ ).
Now let [ f ] ∈ Mod(Σ) − Mod(Hφ ), so that f∗ (∆φ ) 6< ∆φ . Let γ ∈ ∆φ be an element such that (the
conjugacy class of) f∗ (γ ) is not in ∆φ . (In fact, well-defining f∗ (γ ) requires a choice of basepoint
preserving representative homeomorphisms for the mapping class of f , which we make arbitrarily.)
Then φ∗ ( f∗ (γ )) ∈ Fg − {1}, and so by Lemma 5, we can find a prime p and a PSL(2, p)–defining
subgroup N ∈ X(Fg , PSL(2, p)) so that φ∗ ( f∗ (γ )) 6∈ N. Therefore,
f∗ (γ ) 6∈ φ∗−1 (N) ∈ X φ (π1 (Σ), PSL(2, p)),
and hence [ f ] 6∈ stab X φ (π1 (Σ), PSL(2, p)). Since stab X φ (π1 (Σ), PSL(2, p)) is a finite index subgroup containing Mod(Hφ ) (by Lemma 7) and not containing [ f ], and since [ f ] was arbitrary, it
follows that Mod(Hφ ) is separable.
Since Mod0 (Hφ ) < Mod(Hφ ) and since Mod(Hφ ) is separable, it suffices to consider an element
[ f ] ∈ Mod(Hφ ) \ Mod0 (Hφ ), and produce a finite index subgroup of Mod(Σ) containing Mod0 (Hφ )
and not containing [ f ]. For all p, Mod0 (Hφ ) is contained in the subgroup of stab X φ (π1 (Σ), PSL(2, p))
consisting of those mapping classes that act trivially on X φ (π1 (Σ), PSL(2, p)). Since [ f ] 6∈ Mod0 (Hφ ),
Φ∗ ([ f ]) 6= 1 in Out(Fg ). For g ≥ 3, Corollary 4 implies that for some p, Φ∗ ([ f ]) acts nontrivially
on X(Fg , PSL(2, p)). Therefore, [ f ] acts nontrivially on X φ (π1 (Σ), PSL(2, p)), and so the finite index
subgroup G < Mod(Σ) consisting of those mapping classes preserving the subset X φ (π1 (Σ), PSL(2, p))
and acting trivially on this does not contain [ f ], proving that Mod0 (Hφ ) is separable.
Mapping class groups were shown to be residually finite by Grossman as a consequence of the
fact that surface groups are conjugacy separable; see [Gro75].

Quotients of mapping class groups from Out(Fn )

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Theorem 9 (Grossman). Mapping class groups are residually finite.
Residual finiteness of Mod(Σg ) follows immediately from separability of the handlebody subgroups Mod(Hφ ), and the following.
Lemma 10. The intersection of all handlebody subgroups Mod(Hφ ), over all homeomorphisms
φ : Σg → ∂ H is trivial if g ≥ 3, and isomorphic to Z/2Z if g = 2. The intersection of handlebody
subgroups Mod0 (Hφ ) is trivial for all g ≥ 2.
Proof. In [Mas86], Masur proved that the limit set of the handlebody subgroup in the Thurston
boundary of Teichmüller space is a nowhere dense subset. The intersection of all handlebody subgroups is a normal subgroup and so is either finite, or else has limit set equal to the entire Thurston
boundary. By Masur’s result, we must be in the former case, and hence the intersection of handlebody subgroups is finite. But Mod(Σg ) has no nontrivial finite, normal subgroups if g ≥ 3, while
for g = 2, the only nontrivial, finite normal subgroup is the order-two subgroup generated by the
hyperelliptic involution. This proves the first statement. The second follows from the first and the
fact that the hyperelliptic involution of Σ2 induces a nontrivial automorphism of F2 ∼
= π1 (H), for any
homeomorphism φ : Σ2 → H.
Proof of Theorem 9 for Mod(Σg ), with g ≥ 2. An equivalent formulation of residual finiteness is that
the intersection of all finite index subgroups is trivial. Therefore Theorem 8 and Lemma 10 immediately implies the result.

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Khalid Bou-Rabee
Department of Mathematics, CCNY CUNY
E-mail: khalid.math@gmail.com
Christopher Leininger
Department of Mathematics, University of Illinois Urbana-Champaigne
E-mail: c.j.leininger95@gmail.com

