On the profinite rigidity of surface groups and surface words

TL;DR

This paper investigates the profinite rigidity of surface groups, showing they are uniquely determined by their profinite completions, and establishes that surface words form a closed set in the profinite topology.

Contribution

It proves that surface groups are distinguished among limit groups by their profinite completions and that surface words are closed in the profinite topology.

Findings

Surface groups are uniquely identified by their profinite completions.

Surface words form a closed set in the profinite topology.

The results connect surface group properties with profinite topological structures.

Abstract

Surface groups are determined among limit groups by their profinite completions. As a corollary, the set of surface words in a free group is closed in the profinite topology.