Abstract commensurators of right-angled Artin groups and mapping class groups

Consider the mapping class group Mod g,p of a surface Σ g,p of genus g with p punctures, and a finite collection {f1, . . . , fk} of mapping classes, each of which is either a Dehn twist about a simple closed curve or a pseudo-Anosov homeomorphism supported on a connected subsurface. In this paper we prove that for all sufficiently large N, the mapping classes $${\{f_1^N,\ldots,f_k^N\}}$$ generate a right-angled Artin group. The right-angled Artin group which they generate can be determined from the combinatorial topology of the mapping classes themselves. When {f1, . . . , fk} are arbitrary mapping classes, we show that sufficiently large powers of these mapping classes generate a group which embeds in a right-angled Artin group in a controlled way. We establish some analogous results for real and complex hyperbolic manifolds. We also discuss the unsolvability of the isomorphism problem for finitely generated subgroups of Mod g,p , and recover the fact that the isomorphism problem for right-angled Artin groups is solvable. We thus characterize the isomorphism type of many naturally occurring subgroups of Mod g,p .

The group of 2-by-2 matrices with integer entries and determinant $\pm > 1$ can be identified either with the group of outer automorphisms of a rank two free group or with the group of isotopy classes of homeomorphisms of a 2-dimensional torus. Thus this group is the beginning of three natural sequences of groups, namely the general linear groups ${\rm{GL}}(n,\Z)$, the groups of outer automorphisms of free groups of rank $n\geq 2$, and the mapping class groups of closed orientable surfaces of genus $g\geq 1$. Much of the work on mapping class groups and automorphisms of free groups is motivated by the idea that these sequences of groups are strongly analogous. In this article we highlight a few of the most striking similarities and differences between these series of groups and present a list of open problems motivated by this philosophy.

Chapter 12 - Mapping Class Groups

We prove that every RAAG (Right Angled Artin Group) embeds in the group of Hamiltonian symplectomorphisms of every symplectic manifold.

We derive two types of linearity conditions for mapping class groups of orientable surfaces: one for once-punctured surface, and the other for closed surface, respectively. For the once-punctured case, the condition is described in terms of the action of the mapping class group on the deformation space of linear representations of the fundamental group of the corresponding closed surface. For the closed case, the condition is described in terms of the vector space generated by the isotopy classes of essential simple closed curves on the corresponding surface. The latter condition also describes the linearity for the mapping class group of compact orientable surface with boundary, up to center.

We study decorated mapping class groups, i.e., mapping class groups of surfaces with marked points and boundary components, and their behaviour under stabilization maps with respect to the genus, the number of punctures and boundary components. Decorated mapping class groups are non-trivial extensions of the undecorated mapping class group, and the first result states that the extension is homologically trivial when one stabilizes with respect to the genus. The second result implies that one also gets splittings of homology groups when stabilizing with respect to the number of punctures and boundary components.

We prove that if G is a Polish group and A a group admitting a system of generators whose associated length function satisfies: (i) if 0 < k < ω , then l g ( x ) ≤ l g ( x k ) ; (ii) if l g ( y ) < k < ω and x k = y , then x = e , then there exists a subgroup G * of G of size b (the bounding number) such that G * is not embeddable in A. In particular, we prove that the automorphism group of a countable structure cannot be an uncountable right-angled Artin group. This generalizes analogous results for free and free abelian uncountable groups.

Since the 1970's, physicists and mathematicians who study random matrices in the GUE or GOE models are aware of intriguing connections between integrals of such random matrices and enumeration of graphs on surfaces. We establish a new aspect of this theory: for random matrices sampled from the group $\mathcal{U}\left(n\right)$ of unitary matrices. More concretely, we study measures induced by free words on $\mathcal{U}\left(n\right)$. Let $F_{r}$ be the free group on $r$ generators. To sample a random element from $\mathcal{U}\left(n\right)$ according to the measure induced by $w\in F_{r}$, one substitutes the $r$ letters in $w$ by $r$ independent, Haar-random elements from $\mathcal{U}\left(n\right)$. The main theme of this paper is that every moment of this measure is determined by families of pairs $\left(\Sigma,f\right)$, where $\Sigma$ is an orientable surface with boundary, and $f$ is a map from $\Sigma$ to the bouquet of $r$ circles, which sends the boundary components of $\Sigma$ to powers of $w$. A crucial role is then played by Euler characteristics of subgroups of the mapping class group of $\Sigma$. As corollaries, we obtain asymptotic bounds on the moments, we show that the measure on $\mathcal{U}\left(n\right)$ bears information about the number of solutions to the equation $\left[u_{1},v_{1}\right]\cdots\left[u_{g},v_{g}\right]=w$ in the free group, and deduce that one can ``hear'' the stable commutator length of a word through its unitary word measures.

We study stable commutator length on mapping class groups of certain infinite-type surfaces. In particular, we show that stable commutator length defines a continuous function on the commutator subgroups of such infinite-type mapping class groups. We furthermore show that the commutator subgroups are open and closed subgroups and that the abelianizations are finitely generated in many cases. Our results apply to many popular infinite-type surfaces with locally coarsely bounded mapping class groups.

Let (X,≤) be a partially ordered set (in short, a poset). The automorphism group Aut (X,≤) is the group of all permutations g of X such that x ≤ y if and only if xg ≤ yg for all x,y∈X. We say that (X,≤) is sharply transitive if Aut(X,≤) is sharply transitive on X, that is, for x,y∈X there exists a unique g∈Aut(X,≤) with y = xg. Sharply transitive totally ordered sets have been studied by Ohkuma[4, 5], Glass, Gurevich, Holland and Shelah [3] (see also [2] and [6]). Whereas the only countable sharply transitive totally ordered set is the set of integers, there are a great variety of countable sharply transitive posets. Amongst other results, in [1] the author showed that there are countably many non-isomorphic sharply transitive posets whose automorphism group is infinite cyclic (and also gave a full description of those), whereas there are 2N0 non-isomorphic sharply transitive posets whose automorphism group is isomorphic to the additive group of the rational numbers. This suggests also that one should consider the analogous problem for free abelian groups. The purpose of this note is to show that whenever G is a countable free abelian group then there exists a sharply transitive poset whose automorphism group is isomorphic to G, and that there are already 2N0 non-isomorphic sharply transitive posets whose automorphism group is the free abelian group of rank 2.

It is shown that the natural map from the mapping class groups of surfaces to the automorphism groups of free groups induces an infinite loop map on the classifying spaces of the stable groups after plus construction. The proof uses automorphisms of free groups with boundaries which play the role of mapping class groups of surfaces with several boundary components.

Superinjective simplicial maps of complexes of curves and injective homomorphisms of subgroups of mapping class groups

Homological representations of low genus mapping class groups

Let M n M_n be the connect sum of n n copies of S 2 × S 1 S^2 \times S^1 . A classical theorem of Laudenbach says that the mapping class group Mod ⁡ ( M n ) \operatorname {Mod}(M_n) is an extension of Out ⁡ ( F n ) \operatorname {Out}(F_n) by a group ( Z / 2 ) n (\mathbb {Z}/2)^n generated by sphere twists. We prove that this extension splits, so Mod ⁡ ( M n ) \operatorname {Mod}(M_n) is the semidirect product of Out ⁡ ( F n ) \operatorname {Out}(F_n) by ( Z / 2 ) n (\mathbb {Z}/2)^n , which Out ⁡ ( F n ) \operatorname {Out}(F_n) acts on via the dual of the natural surjection Out ⁡ ( F n ) → G L n ( Z / 2 ) \operatorname {Out}(F_n) \rightarrow GL_n(\mathbb {Z}/2) . Our splitting takes Out ⁡ ( F n ) \operatorname {Out}(F_n) to the subgroup of Mod ⁡ ( M n ) \operatorname {Mod}(M_n) consisting of mapping classes that fix the homotopy class of a trivialization of the tangent bundle of M n M_n . Our techniques also simplify various aspects of Laudenbach’s original proof, including the identification of the twist subgroup with ( Z / 2 ) n (\mathbb {Z}/2)^n .

A result of Bridson, Howie, Miller and Short states that if $S$ is a subgroup of type $FP_{n}(\mathbb{Q})$ of the direct product of $n$ limit groups over free groups, then $S$ is virtually the direct product of limit groups over free groups. Furthermore, they characterise finitely presented residually free groups. In this paper these results are generalised to limit groups over Droms right-angled Artin groups. Droms RAAGs are the right-angled Artin groups with the property that all of their finitely generated subgroups are again RAAGs. In addition, we show that the generalised conjugacy problem is solvable for finitely presented groups that are residually a Droms RAAG and that their finitely presentable subgroups are separable.