For finitely presented relatively hyperbolic groups with well-behaved peripheral subgroups, the generalized membership problem is decidable for relatively quasi-convex subgroups.
Algorithms detecting stability and Morseness for finitely generated groups
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abstract
The notions of stable and Morse subgroups of finitely generated groups generalize the concept of a quasiconvex subgroup of a word-hyperbolic group. For a word-hyperbolic group $G$, Kapovich provided a partial algorithm which, on input a finite set $S$ of $G$, halts if $S$ generates a quasiconvex subgroup of $G$ and runs forever otherwise. In this paper, we give various detection and decidability algorithms for stability and Morseness of a finitely generated subgroup of mapping class groups, right-angled Artin groups, toral relatively hyperbolic groups, and finitely generated groups discriminated by a locally quasiconvex torsion-free hyperbolic group (for example, ordinary limit groups).
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2019 1verdicts
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On the generalized membership problem in relatively hyperbolic groups
For finitely presented relatively hyperbolic groups with well-behaved peripheral subgroups, the generalized membership problem is decidable for relatively quasi-convex subgroups.