Stable subgroups and Morse subgroups in mapping class groups

TL;DR

This paper investigates the relationship between stable and Morse subgroups in mapping class groups, proving their equivalence for infinite index subgroups in certain surface groups, thus advancing understanding of subgroup structures.

Contribution

It establishes that in mapping class groups, stable and Morse subgroups of infinite index are equivalent, clarifying subgroup classifications in these groups.

Findings

Stable and Morse subgroups coincide for infinite index subgroups.

Proves equivalence of stability and Morse properties in mapping class groups.

Enhances understanding of subgroup structures in geometric group theory.

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 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 and Genevois. 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.