Stable subgroups and Morse subgroups in mapping class groups

Abstract

For a finitely generated group, there are two recent generalizations of the notion of a quasiconvex subgroup of a word-hyperbolic group, namely a stable subgroup and a Morse or strongly quasiconvex subgroup. Durham and Taylor [DT15] defined stability and proved stability is equivalent to convex cocompactness in mapping class groups. Another natural generalization of quasiconvexity is given by the notion of a Morse or strongly quasiconvex subgroup of a finitely generated group, studied recently by Tran [T17] and Genevois [G17]. In general, a subgroup is stable if and only if the subgroup is Morse and hyperbolic. In this paper, we prove that two properties of being Morse and stable coincide for a subgroup of infinite index in the mapping class group of an oriented, connected, finite type surface with negative Euler characteristic.

Algorithms detecting stability and Morseness for finitely generated groups

“…The equivalence between (1) and (4) shown in [DT15] excludes those two surfaces. The equivalence between (4) and (5) shown in [K19,RST18]. 2.1.…”

Convexity in hierarchically hyperbolic spaces

“…It has been shown that strongly quasiconvex subgroups of the mapping class group [38], right-angled Artin groups with connected defining graph [28; 54], and certain CF S right-angled Coxeter groups (Nguyen and Tran [43]) are either hyperbolic or finiteindex. We give sufficient conditions for a hierarchically hyperbolic space to have the property that all its strongly quasiconvex subsets are either hyperbolic or coarsely cover the entire space; see Proposition 7.2.…”

“…These result have found applications in understanding the cell stabilizers of groups acting on CAT(0) cube complexes -see Groves and Manning [33] -and the splittings of groups over codimension 1 subgroups -see Petrosyan [46]. Using the name Morse instead of strongly quasiconvex, Genevois studied strongly quasiconvex subsets of CAT(0) cube complexes in [28] and Kim studied strongly quasiconvex subgroups of the mapping class groups in [38]. Strongly quasiconvex subgroup that are also hyperbolic were introduced by Durham and Taylor as stable subgroups [25] and have received considerable study; for a sampling see Abbott, Behrstock and Durham [1], Antolín, Mj, Sisto and Taylor [3], Aougab, Durham and Taylor [4], Behrstock [7], and Koberda, Mangahas and Taylor [39].…”

Mapping class groups are quasicubical

“…Quasiconvexity is an important concept in the study of hyperbolic spaces, and there have been various generalisations to larger classes of spaces, many of which coincide for mapping class groups. Indeed, the infinite-index convex cocompact subgroups of MCGpSq coincide with the Morse subgroups [Kim19] and the stable subgroups [DT15]. These quasiconvexity conditions are quite restrictive, and such subgroups are always hyperbolic.…”