Twisted Conjugacy in Houghton's groups

For a fixed $n\ge 2$, the Houghton group $H_n$ consists of bijections of $X_n=\{1,\dots ,n\}\times \mathbb{N}$ 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 $n\ge 2$ and any group $G$ commensurable to $H_n$, $G$ has solvable conjugacy problem.