ABSTRACT COMMENSURATORS: A BRIEF SURVEY
DANIEL STUDENMUND
Contents
1. Introduction
2. Commensuration basics
2.1. Definitions
2.2. Remarks
2.3. Canonical maps
2.4. Universal property
3. Commensurators of lattices and related groups
3.1. Lattices in semisimple Lie groups not PSL2 (R)
3.2. Surface groups
3.3. Free groups and tree lattices
3.4. Mapping class groups
3.5. Outer automorphism group of Fn
3.6. Braid groups
4. Commensurators of other discrete groups
4.1. Surface-by-free
4.2. Thompson’s group F
4.3. Grigorchuk group
4.4. Baumslag-Solitar groups
5. Commensurators of profinite groups
References
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1. Introduction
The abstract commensurator of a group G, denoted Comm(G), is the
group of equivalence classes of isomorphisms φ : H → K, where H and K
are finite index subgroups of G, under composition. The abstract commensurator of G may be thought of as a natural relaxation of the automorphism
group of G, just as the quasi-isometry group of a metric space X is a relaxation of the isometry group of X.
Abstract commensurator groups are generally harder to compute than
automorphism groups, because finite index subgroups have less structure
than the ambient group. Most computations of abstract commensurator
Date: June 6, 2014.
1
2
DANIEL STUDENMUND
groups involve understanding some structure of finite index subgroups, and
many involve the use of rigidity results.
Abstract commensurator groups appear with modest but increasing frequency in the literature. This note presents an overview of the basic properties of abstract commensurator groups, then surveys the known results in
the literature.
The references do not represent an exhaustive list of papers relevant to
the results discussed. My intent is to compile a list of references whose
union contains thorough exposition of every relevant result contained herein
while minimizing the number of purely technical papers. Most references
are original work, a notable exception being Zimmer’s book [Zim84] as a
reference for the theory of semisimple Lie groups.
Acknowledgements: I am grateful to Benson Farb, with whom I had innumerable conversations about abstract commensurators and who suggested
that I write this document. I also thank Alex Lubotzky for stimulating
conversations.
2. Commensuration basics
2.1. Definitions.
Definition 1. Let G be a discrete group. A partial automorphism of G is
a group isomorphism between two finite index subgroups of G.
Suppose φ1 : H1 → K1 and φ2 : H2 → K2 are partial automorphisms of
G. We say that φ1 is equivalent to φ2 , written φ1 ∼ φ2 , if there is some
finite index subgroup H3 of H1 ∩ H2 so that φ1 |H3 = φ2 |H3 . Let [φ] denote
the equivalence class of the partial automorphism φ.
Definition 2. The abstract commensurator of G, denoted Comm(G), is the
set of equivalence classes of partial automorphisms. Elements of Comm(G)
are called commensurations of G.
There is a well-defined composition of commensurations of G; if φ1 :
H1 → K1 and φ2 : H2 → K2 are partial automorphisms, then [φ1 ] ◦ [φ2 ]
is the equivalence class of φ1 ◦ φ2 |φ−1 (H1 ∩K2 ) . The set Comm(G) forms a
2
group under composition.
Commensurations are sometimes called ‘virtual automorphisms,’ and the
abstract commensurator group appears in the literature as the ‘virtual automorphism group.’ We eschew this terminology, as it suggests that commensurations all arise as automorphisms of finite index subgroups. To see that
this does not hold in general, note that any A ∈ GLn (Q) with det(A) 6= 1 induces a commensuration of Zn not equivalent to any automorphism of a finite
index subgroup. See [Odd05] for an example of an element of Comm(π1 (S2 ))
not equivalent to an automorphism of a finite index subgroup.
ABSTRACT COMMENSURATORS: A BRIEF SURVEY
3
2.2. Remarks. If H ≤ G is finite index, then restriction induces an isomorphism Comm(G) ∼
= Comm(H). For example, Comm(F2 ) ∼
= Comm(Fn )
and Comm(π1 (S2 )) ∼
Comm(π
(S
))
for
all
n,
g
≥
2.
More
generally, if
=
1 g
∼
G1 and G2 are abstractly commensurable, then Comm(G1 ) = Comm(G2 ).
Two group G1 and G2 are said to be abstractly commensurable if there are
Hi ≤ Gi with [Gi : Hi ] < ∞ for i = 1, 2 so that H1 ∼
= H2 .
2.2.1. Restricting Domains. Given a commensuration represented by a partial automorphism, one may pass to an equivalent partial automorphism
with nicer domain or range. Let S(G) be the poset of finite index subgroups
of G ordered by reverse inclusion, where H ≤ K if and only if K ⊆ H. This
forms a directed set. It is often useful to identify cofinal subsets of S(G)
consisting of subgroups with particular properties.
Example 3. Let G be a group.
• The subset of finite index normal subgroups of G is cofinal in S(G),
for if [G : H] < ∞ then the kernel of the map G → Sym(G/H)
induced by left translation is a normal subgroup of finite index contained in H.
• If G is finitely generated, then there are finitely many subgroups of a
given index. The intersection of all subgroups with index less than or
equal to any given positive integer n is a characteristic subgroup of
finite index in G. Therefore S(G) contains a cofinal totally ordered
system of characteristic subgroups if G is finitely generated.
Therefore given any [φ] ∈ Comm(G), one may assume without loss of
generality that the domain of φ is normal in G, and characteristic if G is
finitely generated.
2.2.2. Size of Comm(G). If G is finitely generated, then there every finite
index subgroup is finitely generated and there are countably many finite
index subgroups. Since there are countably many isomorphisms between
any two finitely generated groups, it follows that Comm(G) is countable for
finitely generated G.
Comm(G) is typically much larger than Aut(G). For example, Aut(Z) ∼
=
∗
∼
Z/2Z is finite, while Comm(Z) = Q is not even finitely generated. Bartholdi
and Bogopolski have exhibited in [BB10] a number of conditions on G that
guarantee infinite generation of Comm(G). However, for certain groups
Comm(G) is not much larger than Aut(G). If Γ is either an irreducible
nonarithmetic lattice in a nice semisimple Lie group, a sufficiently complex
mapping class group, or Out(Fn ) for n ≥ 4, then [Comm(Γ) : Γ] < ∞.
A. Lubotzky was interested in proving that nontriviality of Comm(G)
is equivalent to G being infinite (see [Man87]). This was proven false by
Mengazzo and Tomkinson in [MT90]. Their counterexamples are uncountable. In contrast, Lubotzky’s conjecture is easily verified for finitely generated groups:
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DANIEL STUDENMUND
Proposition 4. If G is finitely generated and Comm(G) is trivial, then G
is finite.
Proof. Let g1 , . . . , gn be generators for G. Then each map cgi : G → G for
i = 1, . . . , n is trivial in Comm(G). This means that there are finite index
subgroups Gi ≤ G so that gi centralizes Gi for each i. Their intersection
is a central subgroup of G of finite index. Therefore G is virtually abelian,
and hence virtually Zn . But Comm(Zn ) is nontrivial for n ≥ 1, so G must
be finite.
2.2.3. Unique Root Property. A group G satisfies the unique root property if
xn = y n implies x = y for all x, y ∈ G. Examples of such groups include free
groups, free abelian groups, and fundamental groups of closed surfaces. It is
easy to see (see [BB10], [Odd05]) that if G satisfies the unique root property,
and φ1 : H1 → K1 and φ2 : H2 → K2 are equivalent partial automorphisms
of G, then φ1 and φ2 agree on H1 ∩ H2 . It follows that:
Lemma 5. Suppose G satisfies the unique root property, and H ≤ G is of
finite index. Then the natural map Aut(H) → Comm(G) is injective.
Commensurators of groups with the unique root property are then typically quite large. For example:
Proposition 6. Every finite group embeds in Comm(F2 ), where F2 is the
free group on two generators.
Proof. Every finite group embeds in the symmetric group Σn for some n.
The symmetric group Σn embeds in Aut(Fn ) by permuting the generators.
Every Fn is isomorphic to a finite index subgroup of F2 , so Aut(Fn ) embeds
in Comm(F2 ) by the previous lemma.
2.3. Canonical maps. In this section we describe several canonical maps
to and from the abstract commensurator.
Given g ∈ G, let cg : G → G be the conjugation map cg (x) = gxg −1 .
There is a map
ι̂ : G → Comm(G)
given by g 7→ [cg ]. The kernel of ι̂ is the virtual center of G, the collection
of elements of G whose centralizer is finite index in G, denoted by V Z(G).
In the unpublished note [Hil07], it is shown that V Z(G) is locally nilpotent
by locally finite.
Moreover, there is a natural map
Ψ : Aut(G) → Comm(G)
extending ι̂. This map may not be injective. For example, every commensuration of a finite group is trivial. Menegazzo and Robinson [MR87] give
a characterization of groups for which Ψ is trivial. On the other hand, the
map Ψ will be injective if G satisfies the unique root property (see §2.2.3).
ABSTRACT COMMENSURATORS: A BRIEF SURVEY
5
Suppose that G ≤ H. Then there is another natural way to extend ι̂.
The relative commensurator of G in H is
CommH (G) = {h ∈ H | [G : hGh−1 ∩G] < ∞ and [hGh−1 : hGh−1 ∩G] < ∞}.
There is a map
ι : CommH (G) → Comm(G)
extending ι̂, given by h 7→ [ch ]. The kernel of ι is the virtual centralizer of
G in H, defined as
V ZH (G) = {h ∈ H | [G : C(h) ∩ G] < ∞}.
The map ι may be virtually an isomorphism, such as for certain lattices in
semisimple Lie groups and for Thompson’s group F in a certain subgroup
of piecewise linear homeomorphisms of the real line (see §3.1 and §4.2). For
G with trivial virtual center, the map ι realizes Comm(G) as the universal
relative commensurator; see §2.4.
If G is a finitely generated group, then there is a natural map
Comm(G) → QI(G),
which was shown by Kevin Whyte in [FM02a] to be injective. Here QI(G)
is the quasi-isometry group of G, the set of equivalence classes of self quasiisometries of G with respect to some word metric d, where f1 and f2 are
equivalent if there is some D so that d(f1 (x), f2 (x)) ≤ D for all x ∈ G. A
commensuration [φ] gives a quasi-isometry by precomposing with closestpoint projection from G to the domain of φ. In [FM02a], it is shown that
this map is an isomorphism in the case that the map G → QI(G) defined
by g 7→ [cg ] is an injection with finite index image.
2.4. Universal property. In this section we show that every element of
Comm(G) commensurates ι̂(G), and in fact Comm(G) is the universal relative commensurator in the case that G embeds in Comm(G).
Lemma 7. CommComm(G) (ι̂(G)) = Comm(G) for any group G.
Proof. Suppose φ : G1 → G2 is a partial automorphism of G. Note that
φ ◦ cg1 ◦ φ−1 = cφ(g1 ) for any g1 ∈ G1 .
It follows that ι̂(G2 ) ⊆ φ ι̂(G)φ−1 ∩ ι̂(G) and so
[ι̂(G) : φ ι̂(G)φ−1 ∩ ι̂(G)] < ∞.
Because ι̂(G2 ) = φ ι̂(G1 )φ−1 , we have also that
[φ ι̂(G)φ−1 : φ ι̂(G)φ−1 ∩ ι̂(G)] < ∞.
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DANIEL STUDENMUND
Let COM(G) be the category whose objects are inclusions G ,→ H satisfying CommH (G) = H, and whose morphisms are group homomorphisms
H1 → H2 so that the following triangle commutes.
Nn G p
H1
~
/ H2
Proposition 8. Let G be a group with trivial virtual center. Then the natural map ι : G → Comm(G) is the terminal object in the category COM(G).
Proof. The above lemma shows that the natural inclusion G ,→ Comm(G)
gives an object in COM(G). Suppose G ≤ H. We have only to show that
the natural map ι : CommH (G) → Comm(G) described above is the unique
map making the following commute.
Kk G r
y
CommH (G)
ι
$
/ Comm(G)
We know that ι(g) = [cg ] for all g ∈ G. Now suppose h ∈ CommH (G).
There are finite index subgroups G1 , G2 ≤ G so that hG1 h−1 = G2 . Suppose
φ : G3 → G4 is a partial automorphism representing ι(h). Then
[chgh−1 ] = ι(hgh−1 ) = ι(h) ◦ [cg ] ◦ ι(h)−1 = [φ ◦ cg ◦ φ−1 ] = [cφ(g) ]
for all g ∈ G1 ∩ G3 . Since G has trivial virtual center, the assignment
g 7→ [cg ] is an injection. Therefore φ(g1 ) = hgh−1 for all g ∈ G1 ∩ G3 , and
so ι(h) is uniquely determined as an element of Comm(G).
One might attempt to remove the assumption that V Z(G) = 1 from
Proposition 8 by removing the requirement that homomorphisms G → H be
injective in the definition of the category COM(G). However, the terminal
object of this new category is the unique map to the trivial group, so the
obvious modification of Proposition 8 fails.
3. Commensurators of lattices and related groups
3.1. Lattices in semisimple Lie groups not PSL2 (R). The abstract
commensurator group appears in results of Margulis concerning lattices in
semisimple Lie groups, following work of Borel, Mostow, and Prasad. Collectively these results show that the abstract commensurator of an arithmetic
lattice Γ in a ‘sufficiently nice’ simple Lie group G is virtually isomorphic to
a dense subgroup of G. We summarize the situation here. This is explained
by A’Campo and Burger in [AB94]; see also [Stu13]. A good reference for
the background material of this subsection is [Zim84].
ABSTRACT COMMENSURATORS: A BRIEF SURVEY
7
The following theorem is due to Mostow for uniform lattices, was generalized to nonuniform lattices in rank 1 groups by Prasad, and was proven
by Margulis for lattices in higher rank.
Theorem 9 (Strong rigidity, see ). Suppose that G1 , G2 are semisimple Lie
groups with trivial center and no compact factors, with lattices Γ1 ≤ G1 and
Γ2 ≤ G2 . Suppose that Γ1 is irreducible, and G1 6= PSL2 (R). Then any
isomorphism φ : Γ1 → Γ2 extends to a unique isomorphism φ̂ : G1 → G2 .
Corollary 10. Suppose Γ ≤ G is an irreducible lattice in a semisimple Lie
group with trivial center and no compact factors, and G 6= PSL2 (R). Then
Comm(Γ) naturally embeds in Aut(G).
Since G is a semisimple Lie group, its inner automorphism group is finite index in its automorphism group. It follows that the natural map
ι : CommG (Γ) → Comm(Γ) has finite index image. Much is known about
the group CommG (Γ), due to work of Borel and Margulis.
Proposition 11 (Borel). Suppose G is a connected semisimple algebraic
Q-group with trivial center. Then CommG(R) (G(Z)) = G(Q).
Theorem 12 (Margulis). Suppose Γ ≤ G is an irreducible lattice in a connected semisimple Lie group with trivial center and no compact factors. If
[CommG (Γ) : Γ] = ∞, then CommG (Γ) is dense in G and Γ is arithmetic.
Combining the above results, we have the following statement which
rather strikingly characterizes those semisimple lattices arising from arithmetic constructions using purely group theoretic data, namely the criterion
that [Comm(Γ) : Γ] = ∞.
Theorem 13. Suppose that G is a connected semisimple Lie group, G 6=
PSL2 (R), with trivial center and no compact factors. Let Γ ≤ G be an
irreducible lattice. Then either Γ is nonarithmetic and Comm(Γ) is virtually
isomorphic to Γ, or Γ is arithmetic and Comm(Γ) is virtually isomorphic
to a dense subgroup of G.
Margulis’s arithmeticity theorem says that if the real rank of G is at least
2, then every irreducible lattice is arithmetic. This implies that Comm(Γ)
is virtually isomorphic to a dense subgroup of G for all irreducible lattices
Γ in such groups G.
Work due to Schwartz, Farb-Schwartz, and Eskin on quasi-isometric rigidity of lattices in semisimple Lie groups shows the following. (See [Far97] for
an outline of the proof and references.)
Theorem 14. Let Γ be an irreducible nonuniform lattice in a semisimple
Lie group G 6= PSL2 (R) without center. Then
Comm(Γ) ∼
= QI(Γ).
Combining this with the above, we have [QI(Γ) : Comm(Γ)] < ∞ for
Γ ≤ G as in the theorem.
8
DANIEL STUDENMUND
3.2. Surface groups. Now suppose that G = PSL2 (R) and Γ ≤ G is a
lattice. Strong rigidity results of Mostow, Prasad, and Margulis do not
apply, so different methods are required to study Comm(Γ). If Γ is a uniform
lattice, then Γ is virtually isomorphic to π1 (Sg ) by Selberg’s Lemma, where
Sg is a closed surface of genus g ≥ 2. Note that π1 (Sg ) sits as a subgroup of
finite index inside π1 (S2 ), so Comm(π1 (Sg )) ∼
= Comm(π1 (S2 )) for all g ≥ 2.
The group Comm(π1 (S2 )) is the primary object of study by Odden in his
thesis and [Odd05].
Let Γ = π1 (S2 ). Consider the system S(Γ) of finite index subgroups of Γ
partially ordered by inclusion. There is a corresponding directed system of
pointed covers XH of S2 indexed by finite index subgroups H ≤ Γ. Taking
the limit of this system in the category of topological spaces, we define the
universal hyperbolic solenoid
H∞ = lim XH .
←−
H
This space was first studied by Sullivan in [Sul93]. The space H∞ is a comb × Se2 )/Γ, where Γ
b is the profinite
pact Hausdorff space homeomorphic to (Γ
completion of Γ and Se2 is the universal cover of S2 . In this way H∞ is foliated by leaves homeomorphic to H2 , and there is a baseleaf corresponding to
b Odden [Odd05] defines the baseleaf preserving mapping class group
1 ∈ Γ.
ModBLP (H∞ ) and proves
Theorem 15 (Odden). There is a canonical isomorphism
ModBLP (H∞ ) → Comm(Γ).
The proof proceeds via the action of Comm(Γ) on the boundary of the
baseleaf. Fixing an identification of Se2 ∼
= H2 , any commensuration of Γ
induces a quasi-isometry of H2 , and therefore a homeomorphism of ∂H2 . The
map Comm(Γ) → Homeo(∂H2 ) is injective with image CommHomeo(∂H2 ) (Γ).
Odden shows that there is an injective map ModBLP (H∞ ) → Homeo(∂H2 )
with image CommHomeo(∂H2 ) (Γ). The essence of the argument is that any
homeomorphism of H∞ must respect the structure transverse to the foliation
by hyperbolic planes, which constrains the homeomorphisms induced on the
baseleaf.
Now fix an embedding Γ ⊆ PSL2 (R) as a Fuchsian group. The space
b × H2 )/Γ is homeomorphic to H∞ and carries a hyperbolic metric
HΓ∞ = (Γ
on each leaf. An isometry of HΓ∞ is a self-homeomorphism whose restriction
Γ
to any leaf is an isometry. Let Isom+
BLP (H∞ ) denote the group of isometries
that preserve the baseleaf and orientation. Odden notes
Proposition 16. There is a canonical isomorphism
Γ
CommPSL2 (R) (Γ) → Isom+
BLP (H∞ ).
If the embedding Γ ≤ PSL2 (R) is arithmetic then Borel’s result Proposition 11 applies to show that the isometry group of the solenoid HΓ∞ is
ABSTRACT COMMENSURATORS: A BRIEF SURVEY
9
quite large. This is in contrast to the isometry group of a closed hyperbolic
surface, which is always finite.
The system of finite covers XH of S2 induces a codirected system of Teichmüller spaces T (XH ) with inclusion maps T (XH1 ) → T (XH2 ) whenever
XH2 covers XH1 . The colimit T∞ of this system may be interpreted as
the space of transversely locally constant complex structures on the universal hyperbolic solenoid H∞ . This has been studied by Biswas and Nag,
see [BN00].
3.3. Free groups and tree lattices. The free group Fn is the last remaining example of an irreducible lattice in a semisimple Lie group. Though free
groups embed as lattices in PSL2 (R), the lack of strong rigidity means that
methods other than those of §3.1 must be used to study their abstract commensurators.
The abstract commensurator of a free group appears in [BK90, Appendix
B] in the investigation of uniform tree lattices. Let X be a tree, G = Aut(X),
and Γ ≤ G a discrete subgroup with compact quotient. Suppose further that
X is infinite, not virtually a line, and that G leaves no proper subtree of X
invariant. Then Γ is virtually free, so Comm(Γ) ∼
= Comm(F2 ).
Bass and Kulkarni define HE(Γ), the set of Γ-hyperbolic ends of X. This
is naturally identified with the set of commensurability classes of infiniteorder elements of Γ, where γ ∼ γ 0 if there are n, m > 0 so that γ n = γ 0m .
This admits an obvious action of Comm(Γ), and the map Comm(Γ) →
Aut(HE(Γ)) is injective. The space of ends of X has a metric, and the
action of G on the space of ends is continuous with respect to the induced
topology. Bass and Kulkarni ask, is the action of Comm(Γ) on HE(Γ)
continuous with respect to the subspace topology?
Another appearance of Comm(F2 ) is in [PŠ08]. Here Penner and Šarić
define a solenoid analogous to H∞ as the limit of all finite unbranched covers
of the punctured torus. Following Odden’s argument, they identify the baseleaf preserving mapping class group of this solenoid as a certain subgroup
of Comm(F2 ).
3.4. Mapping class groups. Let Sg,n be an orientable surface of genus g
with n punctures. A closed orientable surface of genus g is denoted simply
Sg . The mapping class group Mod(Sg,n ) of Sg,n is the group of isotopy
classes of orientation-preserving homeomorphisms of Sg,n . The extended
mapping class group Mod± (Sg,n ) of Sg,n is the group of all isotopy classes
of homeomorphisms of Sg,n . Note that [Mod± (Sg,n ) : Mod(Sg,n )] = 2.
Harvey [Har79] defined a simplicial complex, the curve complex C(Sg,n ),
on which Mod± (Sg,n ) acts. The vertices of C(Sg,n ) are homotopy classes
of essential simple closed curves, and n vertices form a simplex if their
curves are pairwise disjoint. The action of Mod± (Sg,n ) is analogous to the
action of a lattice in a semisimple Lie group on the associated Tits building.
Ivanov [Iva97] used the curve complex to prove a version of strong rigidity
for mapping class groups:
10
DANIEL STUDENMUND
Theorem 17 (Ivanov). Suppose g ≥ 2 and n ≥ 0. Then the natural map
Mod± (Sg,n ) → Comm(Mod± (Sg,n ))
is an isomorphism.
It is easy to see that V Z(Mod± (Sg,n )) is trivial, so this map is injective.
Ivanov’s main contribution is that every automorphism of C(Sg,n ) is induced
by a mapping class. One then shows that if φ : H → K is a partial automorphism of Mod± (Sg,n ), then it takes powers of Dehn twists to powers of
Dehn twists. Therefore φ induces an automorphism of C(Sg,n ). The induced
mapping class is then shown to conjugate H to K.
In his thesis [Kor96], Korkmaz extends Theorem 17 to the cases g = 0
with n ≥ 5 and g = 1 with n ≥ 3.
One can ask the stronger question of when an injection of a finite index subgroup of Mod± (Sg,n ) is induced by conjugation by some element
of Mod± (Sg,n ). Work of Korkmaz [Kor99], Irmak [Irm04], Bell-Margalit
[BM07], and Behrstock-Margalit [BM06] shows the following:
Theorem 18 (see [BM06]). Suppose 2g + n > 4. Then every injection of a
finite index subgroup of Mod± (Sg,n ) into Mod± (Sg,n ) is induced by an inner
automorphism. If 2g + n ≤ 4, then there is an isomorphism between finite
index subgroups of Mod± (Sg,n ) that is not induced by an inner automorphism.
At first glance there appears to be a contradiction between Theorem 17
and the latter statement of Theorem 18 in the case (g, n) = (2, 0). To resolve
this, we need only show that every inclusion of a finite-index subgroup Γ into
Mod± (S2,0 ) is induced by conjugation on a finite-inde subgroup Γ0 ≤ Γ. To
see this, note that the group Mod± (S2 ) has center isomorphic to Z/2Z, given
by the hyperelliptic involution. Every injection of a finite-index subgroup
Γ ≤ Mod± (S2 ) is induced by conjugation only up to a homomorphism σ :
Γ → Z/2Z. The inclusion of the kernel of σ is induced by conjugation, so
every inclusion of a finite index subgroup is equivalent to one induced by
conjugation. For every pair (g, n) 6= (2, 0) with 2g + n ≤ 4, it is the case
that Comm(Mod± (Sg,n )) 6= Mod± (Sg,n ).
3.4.1. Subgroups of Mod(Sg ). Similar rigidity results have been proven for
many subgroups of mapping class groups using modifications of Ivanov’s
methods. The mapping class group Mod(Sg ) has a natural action on H1 (Sg , Z).
The kernel of this action is the Torelli group I(Sg ). Farb and Ivanov [FI05]
show
Theorem 19. Suppose g ≥ 5. Then the natural map Mod± (Sg ) → Comm(I(Sg ))
is an isomorphism.
The Johnson kernel is the subgroup K(Sg ) of I(Sg ) generated by Dehn
twists about separating curve. In [BM04] and [BM08], Brendle and Margalit
generalize the result of Farb-Ivanov to show
ABSTRACT COMMENSURATORS: A BRIEF SURVEY
11
Theorem 20. Suppose g ≥ 3 and Γ ≤ Mod(Sg ) is isomorphic to either
I(Sg ) or K(Sg ). Then the natural map Mod± (Sg ) → Comm(Γ) is an isomorphism.
This result has been generalized by Kida in [Kid11] to punctured surfaces
Sg,n satisfying one of the following: g = 1 and n ≥ 3; g = 2 and n ≥ 2; or
g ≥ 3.
For closed surfaces of genus g ≥ 4, Bridson-Pettet-Souto have proven
similar rigidity results for many other subgroups. Let Γk be the k th term
in the lower central series of π1 (Sg ), defined by Γ0 = π1 (Sg ) and Γi+1 =
[Γi , π1 (Sg )]. Then Mod± (Sg ) acts by outer automorphisms on π1 (Sg ), and
so acts on π1 (Sg )/Γk for each k; define the Johson kernel Ik (Sg ) to be the
kernel of this action. Then I(Sg ) = I1 (Sg ), and K(Sg ) = I2 (Sg ) by a
theorem of Johnson. Bridson-Pettet-Souto [BPS] prove
Theorem 21. Suppose g ≥ 4. Then the natural map Mod± (Sg ) → Comm(Ik (Sg ))
is an isomorphism for all k.
This generalizes the result of Farb-Ivanov, and partially generalizes the
result of Brendle-Marglit. In fact, Bridson, Pettet, and Souto show that
when g ≥ 4, if Γ ≤ I(Sg ) is normal in Mod± (Sg ) and contains a nonabelian
free group supported on every subsurface Y ⊆ Sg with χ(Y ) ≤ −2, then
Comm(Γ) ∼
= Mod± (Sg ). These conditions are satisfied by every Johnson
kernel, and every term in the lower central series and derived series of I(Sg ).
3.5. Outer automorphism group of Fn . Let Fn be the free group on
n generators. The outer automorphism group Out(Fn ) is a fundamental
object in the study of geometric group theory. There are analogies between
Out(Fn ) and the mapping class group, starting with the Dehn-Nielsen-Baer
theorem Mod± (Sg ) ∼
= Out(π1 (Sg )).
Farb and Handel [FH07] have proven an analog of Ivanov’s theorem:
Theorem 22. Suppose n ≥ 4. Suppose that Γ ≤ Out(Fn ) is a finite index
subgroup and φ : Γ → Out(Fn ) is an injection. Then there is some g ∈
Out(Fn ) so that φ(x) = gxg −1 for all x ∈ Γ.
The methods of [FH07] differ from those in Section 3.4, which are all based
on Ivanov’s curve complex rigidity result. The elements of Out(Fn ) that play
the role of Dehn twists are called elementary automorphisms. However,
there is no geometric object that encodes the algebraic data of elementary
automorphisms so cleanly as the curve complex. Farb-Handel’s work instead
use delicate algebraic arguments to show that φ may be conjugated by some
ψ ∈ Out(Fn ) to virtually fix every elementary automorphism.
3.6. Braid groups. Given an topological spave X, the configuration space
of n distinct ordered points on X is the space Cn (X) = {(x1 , . . . , xn ) ∈ X n |
xi 6= xj }. This space has a natural action of the symmetric group Σn , with
quotient equal to the configuration space of n distinct unordered points on
X.
12
DANIEL STUDENMUND
The braid group on X is Bn (X) = π1 (Cn (X)/Σn ). The braid group
contains the pure braid group, P Bn (X) = π1 (Cn (X)), as a finite index
subgroup. The classical braid group and pure braid group are Bn = Bn (D)
and P Bn = P Bn (D), respectively, where D is the open unit disk.
Braid groups have a close connection with mapping class groups. For a
surface Sg,n with n ≥ 2, the pure mapping class group PMod(Sg,n ) is the
subgroup of Mod(Sg,n ) that takes each puncture to itself. There is a map
PMod(Sg,n ) → Mod(Sg ) by “filling in” each puncture. The kernel of this
map is naturally identified with P Bn (Sg ). Kida and Yamagata [KY11] use
curve complex techniques inspired by Ivanov to prove:
Theorem 23 ( [KY11]). Suppose g 6= 2 and n ≥ 2. Then the natural map
Mod± (Sg,n ) → Comm(Bn (Sg )) is an isomorphism.
The situation for the classical braid groups is slightly different. Charney and Crisp [CC05] noted that the abstract commensurator of any finite
type Artin group, of which the braid group Bn is an example, contains an
infinitely generated free abelian subgroup. The center Z of Bn is infinite
cyclic, and Bn /Z is isomorphic to the finite index subgroup of Mod(S0,n+1 )
of elements fixing a single given puncture. Charney and Crisp then remark
that a corollary of Ivanov and Korkmaz’s work is that Comm(Bn /Z) ∼
=
Mod± (S0,n+1 ).
In [LM06], Leininger and Margalit show
Theorem 24. Suppose n ≥ 4. Then
Comm(Bn ) ∼
= Mod± (S0,n+1 ) n Q× n Q∞ .
Leininger and Margalit’s proof begins with the observation that the center
of Bn virtually splits off as a direct product factor. More precisely, Bn
contains a finite index subgroup that is isomorphic to a direct product G =
Ĝ × Z, where Z ∼
= Z is the center of Bn and Ĝ is isomorphic to a finite index
subgroup of Bn /Z. Since the center of every finite index subgroup of G is
equal to its intersection with Z, the center Z is virtually preserved by every
commensuration. Thus there is a map
Comm(Bn ) ∼
= Comm(G) → Comm(G/Z) ∼
= Mod± (S0,n+1 ).
This map is easily seen to be surjective and the induced short exact sequence
splits. The subfactor of Q× = GL1 (Q) in Theorem 24 is the induced group
of commensurations of Z, and the remaining subfactor Q∞ is the group of
virtual homomorphisms Ĝ → Z.
Leininger and Margalit also remark that their methods extend more generally to describe commensurators of central extensions of groups G with
the property that every finite index subgroup of G is centerless. These comments are the first extending methods for computing automorphism groups
of extensions to methods for abstract commensurators of extensions.
ABSTRACT COMMENSURATORS: A BRIEF SURVEY
13
4. Commensurators of other discrete groups
4.1. Surface-by-free. A Schottky subgroup of Isom(Hn ) is a finitely generated free subgroup whose action on the convex hull of its limit set in ∂Hn
is cocompact. By analogy, a Schottky subgroup H of Mod(Sg ) is a free subgroup of pseudo-Anosov mapping classes whose action on the weak convex
hull of its limit set on the boundary of T (Sg ) is cocompact. To any such H
there is an associated group ΓH satisfying the exact sequence
1 → π1 (Sg ) → ΓH → H → 1.
These have been shown in [FM02b] to be precisely those surface-by-free
groups that are word hyperbolic. In [FM02a], Farb and Mosher show that
such groups ΓH are extremely rigid:
Theorem 25. Suppose H ≤ Mod(Sg ) is Schottky and let ΓH be as above.
Then the natural map ΓH → QI(ΓH ) is an inclusion with finite index image.
Moreover, the natural map Comm(ΓH ) → QI(ΓH ) is an isomorphism, and
these groups can be computed explicitly.
The proof given is heavily coarse-geometric. Farb and Mosher construct
a space XH on which ΓH acts cocompactly, then analyze the dynamics of
quasi-isometries on XH to show that QI(ΓH ) is isomorphic to a certain group
ΓN . The group ΓN is constructed by first taking O the minimal orbifold
subcover of S for which H descends isomorphically to a subgroup of Mod(O).
Letting N = CommMod(O) (H), the group ΓN is an extension of N by π1 (O).
The computation of Comm(ΓH ) is a general consequence of the strength
of this rigidity. Farb and Mosher show that given any finitely generated Γ, if
Γ → QI(Γ) is an inclusion with finite index image, then Comm(Γ) ∼
= QI(Γ).
There are examples in which either Γ ≤ Comm(Γ) or Comm(Γ) ≤ QI(Γ)
may be infinite index. Suppose Λ is a torsion-free lattice in Isom(H3 ). If Λ
is nonuniform and arithmetic, then Comm(Λ) ∼
= QI(Λ) both contain Λ in
infinite index. If Λ is nonarithmetic and uniform then Comm(Γ) is virtually
isomorphic to Γ while QI(Γ) ∼
= QI(H3 ) is much larger.
4.2. Thompson’s group F . Let PL2 (R) denote the set of homeomorphisms f : R → R satisfying:
(1)
(2)
(3)
(4)
f is piecewise linear,
f 0 is discontinuous at a discrete subset of Z[ 21 ],
slopes of f are integral powers of 2, and
f (Z[ 12 ]) ⊆ Z[ 21 ].
Let PL+
2 (R) ≤ PL2 (R) be the orientation-preserving subgroup. Thompson’s
group F may be defined as the group consisting of f ∈ PL+
2 (R) so that
(5) there is some M > 0 and i, j ∈ Z so that f (x) = x + i for all x > |M |
and f (x) = x + j for all x < − |M |.
Burillo, Cleary, and Röver prove in [BCR08]:
14
DANIEL STUDENMUND
Theorem 26. Suppose F and PL2 (R) are defined as above. Then
Comm(F ) ∼
= CommPL2 (R) (F ).
The latter consists of f ∈ PL2 (R) so that there is some M > 0 and p, p0 , q, q 0 ∈
Z satisfying f (t + p) = f (t) + q for all t > |M | and f (t + p0 ) = f (t) + q 0 for
t < − |M |.
Burillo-Cleary-Röver first use that [F, F ] is a simple group to show that
[H, H] = [F, F ] for any H ≤ F of finite index. A general rigidity statement
about isomorphisms between certain groups acting on dense linear orders,
due to McCleary-Rubin, gives a map Ψ : Comm(F ) → Homeo(R). Since
the above shows that [F, F ] is preserved by every partial automorphism, the
image of Ψ is in NHomeo(R) ([F, F ]) ∼
= PL2 (R). This gives the isomorphism
Comm(F ) ∼
= CommPL2 (R) (F ).
Let Comm+ (F ) denote the group of orientation-preserving commensurations of F . Burillo-Cleary-Röver note that the assignment of f ∈ CommPL+ (R) (F )
2
to the pair (p/q, p0 /q 0 ) ∈ Q×Q, where p, p0 , q, q 0 ∈ Z are defined as in the theorem, gives a surjective homomorphism Comm+ (F ) → Q∗ × Q∗ . Therefore
Comm(F ) is infinitely generated.
4.3. Grigorchuk group. Röver [Röv02] has studied commensurators of
some groups acting on rooted trees. Suppose T is a rooted tree with valence
at least 2 at the root, at least 3 at each other vertex, and so that all vertices
at distance n from the root have the same valence. A subgroup G ≤ Aut(T )
is nearly level transitive if there is some K > 0 so that for each n > 0, there
are at most K orbits of G in the set of vertices of distance n from the root.
Say G is weakly branch if for each vertex v there is some nontrivial g ∈ G
so that g is nontrivial only on vertices w so that v lies on the geodesic from
the root to w. Röver proves a rigidity theorem for such groups:
Theorem 27 ( [Röv02]). Suppose T is as above and G ≤ Aut(T ) is nearly
level transitive and weakly branch. Then Comm(G) ∼
= CommHomeo(∂T ) (G).
Because the homeomorphism group of the boundary of T is quite large,
in applications one wishes to understand the latter group more precisely.
Röver proves a strengthening for regular trees with the following corollary.
Theorem 28. Let G be the Grigorchuk 2-group. Then Comm(G) is a
finitely presented infinite simple group generated by G and Thompson’s group
V.
Thompson’s group V appears because G is commensurable with G × G,
and so homeomorphisms of ∂T induced by swapping subtrees at different
levels induce commensurations. Röver in fact proves the stronger statement that if H is commensurable with its own nth product, then Comm(H)
contains a Higman-Thompson group Gn,1 .
ABSTRACT COMMENSURATORS: A BRIEF SURVEY
15
4.4. Baumslag-Solitar groups. The Baumslag-Solitar groups may be defined by the presentation
BS(m, n) = a, b | abm a−1 = bn .
The groups BS(1, n) are important examples of non-polycyclic solvable
groups. Bogopolski [Bog10] has shown:
Theorem 29. Comm(BS(1, n)) ∼
= Q o Q∗ .
The proof proceeds first by understanding very precisely the structure of
finite index subgroups. In particular, every finite index subgroup of BS(1, n)
is isomorphic to BS(1, nk ) for some k ≥ 1. The proof of the theorem then
relies on careful understanding of the automorphism groups of BS(1, nk ).
Roughly speaking, the factor of Q∗ comes from stretching in the hai direction, while the factor of Q comes from shearing in the hbi direction.
5. Commensurators of profinite groups
Barnea, Ershov, and Weigel [BEW11] have developed the theory of abstract commensurators in the context of profinite groups. They define the
commensurator of a profinite group G as all equivalence classes of topological isomorphisms between open subgroups of G, where two isomorphisms are
equivalent if they agree on some open subgroup. We will use Commp to denote the profinite commensurator. A corollary of work of Nikolov and Segal
is that Commp (G) ∼
= Comm(G) whenever G is a profinite group containing
a dense finitely-generated subgroup.
An envelope for a profinite group G is a topological group L and an embedding of G in L as an open subgroup. If V Z(G) = 1, then Commp (G)
is an envelope for G under the natural embedding ιG : G → Commp (G),
with respect to a natural topology on Commp (G) called the strong topology. Moreover, it is universal with respect to this property; this embedding
factors through every other envelope of G.
There are a number of connections between the abstract commensurator
and strong (Mostow) rigidity. For example, in [BEW11] the following is
stated as a corollary of Pink’s work on strong rigidity of open compact
subgroups of certain algebraic groups. (See also [Ers10].)
Theorem 30. Let F be a non-archimedean local field, let G be an absolutely simple simply connected algebraic group over F , and G an open
compact subgroup of G(F ). Then Commp (G) is canonically isomorphic to
(Aut G)(F ) o Aut(F ).
There are a number of other results on commensurators of profinite groups
in [BEW11]. They consider commensurations of the profinite completion of
the first Grigorchuk group, of the Nottingham group, and of absolute Galois
groups.
If F is a finite field, then the Nottingham group N (F ) is defined as
N (F ) = {φ ∈ AutF (F [[t]]) | φ(t) ≡ t mod t2 F [[t]]}.
16
DANIEL STUDENMUND
Theorem 31 (Ershov, [Ers10]). Let p ≥ 5 be prime and N = N (Fp ). Then
the natural map Aut(N ) → Commp (N ) is an isomorphism.
The main tools in the proof of this fact are the correspondence between
subgroups of N and subalgebras of the graded Lie algebra of N , and the
notion of Hausdorff dimension.
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