Twisted Conjugacy in Houghton's groups

TL;DR

This paper proves that all groups commensurable with Houghton's groups have a solvable conjugacy problem, extending known results about Houghton's groups to their finite extensions and subgroups.

Contribution

It establishes that the solvable conjugacy problem property is preserved under commensurability for Houghton's groups, a stronger result than previously known.

Findings

Houghton's groups have solvable conjugacy problem.

Any group commensurable with Houghton's groups also has solvable conjugacy problem.

The result extends the class of groups known to have solvable conjugacy problem.

Abstract

For a fixed $n\ge2$, the Houghton group $H_n$ consists of bijections of $X_n=\{1,\ldots,n\} \times \mathbb{N}$ that are `eventually translations' of each copy of $\mathbb{N}$. The Houghton groups have been shown to have solvable conjugacy problem. In general solvable conjugacy problem does not imply that all finite extensions and finite index subgroups have solvable conjugacy problem. Our main theorem is that a stronger result holds: for any $n\ge2$ and any group $G$ commensurable to $H_n$, $G$ has solvable conjugacy problem.