Isomorphism versus commensurability for a class of finitely presented groups

TL;DR

This paper constructs classes of finitely presented groups demonstrating contrasting solvability of isomorphism and commensurability problems, highlighting novel complexity behaviors in group theory.

Contribution

It provides the first examples where the isomorphism problem is solvable but the commensurability problem is unsolvable, and vice versa.

Findings

Constructed groups with solvable isomorphism but unsolvable commensurability problems.

Constructed groups with solvable commensurability but unsolvable isomorphism problems.

First examples of such contrasting complexity behaviors in finitely presented groups.

Abstract

We construct a class of finitely presented groups where the isomorphism problem is solvable but the commensurability problem is unsolvable. Conversely, we construct a class of finitely presented groups within which the commensurability problem is solvable but the isomorphism problem is unsolvable. These are first examples of such a contrastive complexity behaviour with respect to the isomorphism problem.