# On the generalized membership problem in relatively hyperbolic groups

## TL;DR

This paper proves that the generalized membership problem is decidable for certain subgroups within relatively hyperbolic groups, especially those with toral peripheral structures, under mild conditions.

## Contribution

It establishes the decidability of the generalized membership problem for relatively quasi-convex subgroups in finitely presented relatively hyperbolic groups, broadening understanding in geometric group theory.

## Key findings

- Decidability of the generalized membership problem is proven.
- Results apply to toral relatively hyperbolic groups.
- Conditions on peripheral structures are mild and widely satisfied.

## Abstract

The aim of this short note is to provide a proof of the decidability of the generalized membership problem for relatively quasi-convex subgroups of finitely presented relatively hyperbolic groups, under some reasonably mild conditions on the peripheral structure of these groups. These hypotheses are satisfied, in particular, by toral relatively hyperbolic groups.