Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups

Abstract

Consider the mapping class group Modg,p of a surface Σg,p of genus g with p punctures, and a finite collection {f1, . . . , f k } 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 N 1 , . . . , f N k } 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, . . . , f k } 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 Modg,p, and prove that the isomorphism problem for right-angled Artin groups is solvable. We thus characterize the isomorphism type of many naturally occurring subgroups of Modg,p.2010 Mathematics Subject Classification. Primary 37E30; Secondary 20F36, 05C60.

The geometry of right-angled Artin subgroups of mapping class groups

“…In [25], Koberda observes that Mod(S g ) is not commensurable with a right-angled Artin group if g ≥ 3 (in fact, he proves the stronger statement that Mod(S g ) cannot virtually embed in a right-angled Artin group). This is also true for genus 2 as the following shows.…”

“…, f n ∈ Mod(S), we cannot expect that they generate a free group upon raising to sufficiently high powers. However, Koberda [25] has recently proven that the powers do generate a right-angled Artin group; see also [9,12,8] for partial results in this direction.…”

The geometry of purely loxodromic subgroups of right-angled Artin groups

“…Rightangled Artin groups also play a key role in the study of three-manifold topology, culminating in Agol's resolution of the virtual Haken conjecture [Ago13, KM12,Wis11]. Right-angled Artin groups are also a prototypical class of CAT(0) groups, and have figured importantly in the study of mapping class groups of surfaces [CW04,CLM12,Kob12,KK13,KK14b,MT13].…”

On the subgroups of right-angled Artin groups and mapping class groups

“…With more care, one can arrange for the embedding of the RAAG to lie in the Torelli subgroup T (S) < Mod(S) (cf. Koberda [22]); so the undecidability phenomena described above are already present among the finitely presented subgroups of T (S).…”

“…One such setting is that of surface automorphisms: if two simple closed curves on a surface are disjoint, then the Dehn twists in those curves commute, but if one has a set curves, no pair of which can be homotoped off each other, then high powers of the twists in those curves freely generate a free group. (Significantly sharper results of this sort are proved in [22] and [12].) It follows that any RAAG can be embedded in the mapping class group of any surface S of sufficiently high genus: it suffices that the dual of the graph defining A can be embedded in S. (This is explained by Crisp and Wiest in [14]; I do not know if it appears in the literature earlier.…”