Lattices vs mapping class groups

This post discusses similarities and differences between mapping class groups and lattices in Lie groups, and assumes more background in GGT and related areas than some other posts.

The mapping class group (MCG) of a surface is a very mysterious object that has captured the attention of a great many mathematicians for decades. There is a developed analogy between MCGs, $\rm{Out}(F_n)$, and $\rm{GL}_n(\mathbb{Z})$ which is 'sometimes derailed by inconvenient facts'. Part of this philosophy is that MCGs should share some features with rank 1 lattices, and some features with higher rank lattices. The former are hyperbolic/relatively hyperbolic, while the latter are famously extremely rigid in basically any possible sense. In this post I'd like to point out some interesting cases where the boundary lies in where the MCG resembles the rank one case more.
 
Our starting point is the curve complex, which is a simplicial complex associated to a compact surface. Its vertices correspond to homotopy classes of essential simple closed curves, and an edge connects two vertices if their corresponding curves can be realized disjointly on the surface. Here already we can state a first striking result: the curve complex is $\delta$-hyperbolic, for a $\delta$ independent of the underlying surface! This is due independently to a number of people. See Sisto's blog for an exposition of the short proof of Hensel--Przytycki--Webb, which yields I think the best hyperbolicity constant, as well as an even shorter proof due to Sisto-Przytycki. 

However, the curve graph is not locally finite/compact, and the action of \( \operatorname{MCG}(S) \) on the curve graph is not proper; the stabilizer of a vertex is isomorphic to a copy of \( \operatorname{MCG}(S') \) for some smaller surface \( S' \).  Indeed, the mapping class group is not hyperbolic or $\rm{CAT}(0)$, and one can't pass to a finite index subgroup to increase the injectivity radius as when dealing with hyperbolic groups. Nevertheless, there are some things which can be recovered. For instance, MCGs are weakly relatively hyperbolic, i.e. relatively hyperbolic but without fineness/BCP. I know very few theorems which work specifically for this class, so let's see something more interesting. 

Definition: An element \( a \in \operatorname{MCG}(S) \) is reducible if it permutes some finite set of isotopy classes of disjoint, non-parallel essential simple closed curves. It turns out that an element \( a \in \operatorname{MCG}(S) \) has a finite orbit in \( \mathcal{C}(S) \) if and only if \( a \) is either finite order or reducible. An element which is neither finite order nor reducible is said to be pseudo-Anosov.

Theorem 1 (Masur-Minsky): Let \( a \in \operatorname{MCG}(S) \) be pseudo-Anosov. Then every orbit of \( a \) on \( \mathcal{C}(S) \) is a quasigeodesic.

In particular, every pseudo-Anosov element has a positive translation length \( \tau(a) \).

Theorem 2 (Bowditch): Let \( S \) be a closed, orientable surface of genus at least 2. Then there is a constant \( C(S) \) such that for every pseudo-Anosov \( a \in \operatorname{MCG}(S) \), there is \( n \leq C \) such that \( a^n \) fixes some bi-infinite geodesic axis \( l_a \) and acts on it by translation.

 In the original papers of Masur-Minsky, a lot of technical effort goes into recovering the hyperbolic-like behaviour, but (fortunately) many of the results have been reproved and generalised since then. One such result concerns the finiteness of an important subclass of geodesics between any two points, which Bowditch refine this finiteness result. 

Definition:   A geodesic in the curve complex consists of a sequence of simplices \( A_0, A_1, \ldots, A_n \) with the property that for each \( i, j \), any two curves \( a \in A_i \) and \( b \in A_j \) are at distance \( |i - j| \) apart. Such a geodesic is called tight if for each sequence of vertices \( A_{i-1}, A_i, A_{i+1} \), each element of \( A_i \) is a boundary curve of the connected subsurface filled by \( A_{i-1} \) and \( A_{i+1} \).

 This is slightly weaker than the original definition of tight given by Masur and Minsky, where it is required that \( A_i \) consists of all non-peripheral boundary curves of the subsurface filled by \( A_{i-1} \) and \( A_{i+1} \). Given any two vertices \( a, b \) of the curve complex of a given surface, let \( G(a, b) \) denote the union of all tight geodesics between \( a \) and \( b \).

Bowditch proves the following quantitative finiteness result concerning tight geodesics: 

Theorem 3: Let \( k \) be any constant larger than the hyperbolicity constant for the curve complex of \( S \). Then, there exists a constant \( K \) depending only on the topological type of the surface, so that for any two curves \( a, b \) on that surface and any vertex \( c \in G(a, b) \) there are at most \( K \) vertices in the intersection of \( G(a, b) \) with the ball of radius \( k \) around \( c \).

To prove this, Bowditch analyses hyperbolic 3-manifolds, homotopy equivalent to $S$, in which the homotopy classes of curves corresponding to $a$ and $b$ are realised by “short” closed geodesics.

Recall that the action of a group \( G \) is called acylindrical if for all \( r > 0 \), there exist \( R, N \geq 0 \), such that for all vertices \( a, b \) with \( d(a, b) \geq R \) there are at most \( N \) distinct elements \( g \in G \) such that \( d(a, ga) \leq r \) and \( d(b, gb) \leq r \). Part of the interest in this property is that acylindrical actions on hyperbolic metric spaces implies something called 'weak proper discontinuity' or WPD, which was introduced by Bestvina--Fujiwara as an attempt to recover some sort of properness. They show that a group $G$ with a WPD action has the property that the kernel \( Q(G) \) of the comparison map from second bounded cohomology to ordinary group cohomology is infinite dimensional. Bowditch applies his finiteness results to prove: 

Theorem 4: For any surface \( S \) with \( 3(\text{genus}) + \#(\text{punctures})> 4 \), the action of the mapping class group of \( S \) on the curve complex of \( S \) is acylindrical. 

and hence recovering

Theorem 5 (Bestvina-Fujiwara): Let \( G \) be a subgroup of \( \operatorname{MCG}(S) \) which is not virtually abelian. Then the dimension of \( Q(G) \) is infinite.

In particular, if \( \phi \) is pseudo-Anosov, and \( p, q \) are sufficiently far apart on an axis for \( \phi \), only finitely many elements of \( \operatorname{MCG}(S) \) move both \( p \) and \( q \) a bounded distance. This is enough to show that every pseudo-Anosov element either has a (bounded) power conjugate to its inverse, or has positive scl. The estimates can be made uniform using Bowditch's result.

In fact, this notion of WPD inspired a whole class of groups called acylindrically hyperbolic groups. It is difficult to prove things about the entire class, since for any group $G$, $G * \mathbb{Z}$ is acylindrically hyperbolic, as witnessed by a generator of the $\mathbb{Z}$. Nevertheless, this is still a generalisation of hyperbolicity, and is opposed to the higher rank case by a result of Burger-Monod that  says an irreducible lattice $\Gamma$ in a higher rank connected semisimple group $G$ with finite centre has the property that every homogeneous quasimorphism $\Gamma \to \R$ is $\equiv 0$.

Morse elements

Further important results were reproved by Drutu-Mozes-Sapir, who studied the relation between divergence functions, Morse quasi-geodesics and cut points in asymptotic cones. We recall below what these notions are. As is customary in geometric group theory, we say that \(f \cong g\) if and only if there exists \(C\) such that, for all \(n\),
\[
C^{-1}f(n/C) - C(n + 1) \leq g(n) \leq Cf(Cn) + C(n + 1).
\]

Let \( X \) be a length space. The divergence function of \( X \) measures how much distances increase when balls are removed. There are many equivalent definitions, which the authors exploit to prove things more easily, but here is one. 

Definition: Let \(\delta\) be a positive number less than 1; for any three points \(a, b, c\) in \(X\) let \( \operatorname{div}(a, b, c; \delta) \) be the distance between \(a\) and \(b\) in the space \(X \setminus B(c, \delta r)\) where \(r = \min(d(c, a), d(c, b))\). (In practice one removes a ball of radius \(\delta r - \gamma\), where \(\gamma\) is a fixed constant, to avoid pathologies.) The divergence function \(\operatorname{Div}(n; \delta)\) is then the supremum of the \(\operatorname{div}(a, b, c; \delta)\) where \(a\) and \(b\) are at distance less than \(d\). 

Various restrictions can be made on the choice of \(c\) and it is shown that it leads to the same (equivalence class of) function.

For \(\delta\) small enough, the equivalence class of the divergence function does not change and becomes a quasi-isometry invariant. A function is said to be linear if it is in the equivalence class of the zero function, and superlinear otherwise.

A Morse quasi-geodesic \(p\) in \(X\) is a quasi-geodesic which satisfies the conclusion of the celebrated Morse lemma in hyperbolic metric spaces: every quasi-geodesic having endpoints on \(p\) stays within bounded distance of \(p\) (the involved constant depending only on the quasi-geodesic constants). In a finitely generated group, an element \(g\) is a Morse element if \((g^n)_{n \in \mathbb{Z}}\) is a Morse quasi-geodesic.

Finally, let us recall what are asymptotic cones in the case where \(X = \Gamma\) is a finitely generated group with its (left-invariant) word metric \(| \cdot |\). Fix a ultrafilter \(\omega\) and a sequence \((d_n)\) converging to infinity: \(\lim_{\omega} d_n = +\infty\). The asymptotic cone \(\mathcal{C} = \operatorname{Con}^\omega(\Gamma, (d_n))\) is the quotient of the group \(\{( \gamma_n) \in \Gamma^{\mathbb{N}} \mid \lim_{\omega} |\gamma_n| / d_n < +\infty\}\) by the subgroup \(\{( \gamma_n) \mid \lim_{\omega} |\gamma_n| / d_n = 0\}\). It is a homogeneous, connected, metric space:
\[
d((\gamma_n), (\gamma_n')) = \lim_{\omega} |(\gamma_n)^{-1}\gamma_n'| / d_n.
\]
A point \(c\) in \(\mathcal{C}\) is a cut point if \(\mathcal{C} \setminus \{c\}\) is not connected. It is a slightly ridiculous fact that there are groups whose number of non-homeomorphic asymptotic cones depends on the continuum hypothesis.

Drutu-Mozes-Sapir then show the following relationship between these objects:

Lemma 6:

1. the existence of an acylindrical action implies the existence of many Morse elements.
2. If \(g\) is a Morse element in \(\Gamma\) then \(g\) remains a Morse element in every subgroup of \(\Gamma\) containing \(g\).
3. A Morse quasi-geodesic implies the existence of cut points in every asymptotic cone. 
4. \(\Gamma\) has a cut point in one asymptotic cone if and only if the divergence function is superlinear. 

The main result of their paper is that for group actions on a family of hyperbolic graphs resembling the curve complex, called Bowditch graphs, loxodromic elements are Morse. Let us define Bowditch’s graphs.

Let \( G \) be a \(\delta\)-hyperbolic graph. For any two vertices \( a, b \), choose a non-empty set of geodesics connecting \( a, b \) and call these geodesics tight.

For every \( r \geq 0 \), we denote by \( T(a, b; r) \) the union of all the points on all tight geodesics connecting a point in the ball \( B(a, r) \) with a point in \( B(b, r) \).


Definition: We say that a hyperbolic graph \( G \) equipped with a collection of tight geodesics as above is a Bowditch graph if it satisfies the following conditions:

1.  Every subpath of a tight geodesic is tight. 
2. For every \( r > 0 \) there exist \( m = m(r) \) and \( k = k(r) \) such that for every three vertices \( a, b, c \) in \( G \) with \(\operatorname{dist}(c, \{a, b\}) \geq k\), the set  \[F_c(a, b; r) = B(c, r) \cap T(a, b; r)\] contains at most \( m \) points.

Theorem 7 (Drutu-Mozes-Sapir): Let \( G \) be a finitely generated group acting tightly and acylindrically on a Bowditch graph \( \mathcal{G} \). Then every loxodromic element of \( G \) is Morse. In particular, if \( G \) has lozodromic elements, then every asymptotic cone of \( G \) has cut-points.

The authors use this and Bowditch's earlier result to recover a theorem of Behrstock that asymptotic cones of mapping class groups have cut-points, and for every pseudo-Anosov mapping class \( g \), the sequence \((g^n)_{n \in \mathbb{Z}}\) is a Morse quasi-geodesic.

Applications to subgroups of mapping class groups

One of the most important results about subgroups of the mapping class groups is the Tits alternative proved by McCarthy and Ivanov: every subgroup of a mapping class group \(\mathcal{MCG}(S)\) either contains a free non-abelian subgroup or it contains a free abelian subgroup of rank at most \(\xi(S)\) and of index at most \(N=N(S)\). Thus every subgroup of a mapping class group not containing a free non-abelian subgroup is virtually abelian. A similar result is recovered:

Theorem 8: If\(H\) is a subgroup of the mapping class group \(\mathcal{MCG}(S)\) (where \(S\) is a surface with possible punctures) that does not have Morse elements, then \(H\) stabilizes a (finite) collection of pairwise disjoint simple closed curves on \(S\).

We know from above that the curve complex of a surface \(S\) is a Bowditch graph. By Lemma 6, \(H\) cannot contain any loxodromic elements for the action of \(\mathcal{MCG}(S)\) on the curve complex of \(S\). Thus some power of every element in \(H\) must fix a curve on \(S\). Ivanov and McCarthy show that in this case \(H\) stabilizes a collection of pairwise disjoint simple closed curves on \(S\).

This recovers the result that virtually soluble subgroups must be virtually abelian. Indeed, non-virtually cyclic soluble groups do not have cut-points in their asymptotic cones, so if \(H\) is a soluble non-virtually cyclic subgroup of \(\mathcal{MCG}(S)\), then it must stabilize a collection of pairwise disjoint simple closed curves on \(S\). Hence up to a finite index, it must fix a curve \(\gamma\) on \(S\). Then up to finite index, \(H\) is a subgroup of the direct product of the cyclic subgroup generated by the Dehn twist about \(\gamma\) and the restriction of \(H\) onto \(S\setminus\gamma\), we get a soluble subgroup of the mapping class group of a surface of smaller complexity (the complexity of \(S\) is, by definition, \(3g+p-3\) where \(g\) is the genus, \(p\) is the number of punctures). Using an induction on the complexity, we deduce that up to a finite index, \(H\) is in the subgroup generated by Dehn twists about a collection of pairwise disjoint simple closed curves, so \(H\) is virtually abelian (in fact the rank of the free abelian subgroup of \(H\) does not exceed \(3g+p-3\), and the index does not exceed the maximal size of a finite subgroup of \(\mathcal{MCG}(S)\)).

Furthermore, Theorem 8 implies a superrigidity theorem of Farb-Kaimanovich-Masur: the mapping class group of a surface does not contain lattices of semi-simple Lie groups of higher ranks. (This was originally proved by considering measures on Thurston's compactification on Teichmuller space, and their methods also show that any map to the braid group has finite image.)

For irreducible non-uniform lattices this is so because such lattices contain distorted cyclic subgroups, while Farb, Lubotzky and Minsky proved that the mapping class groups do not. On the other hand, uniform higher rank lattices do not have cut-points in their asymptotic cones. 

This is because an asymptotic cone of a uniform lattice \(\Gamma\) in a Lie group \(L\) is bi-Lipschitz equivalent to an asymptotic cone of \(L\). If \(L\) is semi-simple of higher rank and non-compact type, then every asymptotic cone of \(L\) is a Euclidean building of rank \(\geq 2\) and every two points in the asymptotic cone belong to a 2-dimensional flat. Hence there are no cut-points in the asymptotic cones of \(L\). (The authors also conjecture that all non-uniform lattices have linear divergence.)

By theorem 8, if such a lattice is a subgroup of a mapping class group, then it must stabilize a multi-curve. Suppose that such a lattice \(\Gamma\) stabilizes a curve on \(S\). Then a finite index subgroup of \(\Gamma\) would have a homomorphism with infinite image into the mapping class group of a surface of smaller complexity (the surface cut along the multi-curve). By Selberg's theorem, we can assume that \(\Gamma\) is torsion-free. By Margulis' normal subgroup theorem, the homomorphism must be injective, and we can proceed by induction on the complexity of \(S\). When the complexity is zero, i.e. when the surface is the sphere with three holes, the mapping class group is finite.

In fact, the paper of Farb, Lubotzky and Minsky cited above is called "Rank one phenomena for mapping class groups", and they prove other rank 1 phenomena. The one I find most interesting is that mapping class groups, unlike higher rank lattices, aren't boundedly generated, hence neither are the profinite and pro-$p$ completions. This is really a stark contrast: the $p$-congruence subgroups $\mathrm{SL}_n(\mathbb{Z})$ have uniformly powerful pro-$p$ completions.