Isomorphisms and commensurability of surface Houghton groups

Javier Aramayona, George Domat, Christopher J. Leininger

Abstract

We classify surface Houghton groups, as well as their pure subgroups, up to isomorphism, commensurability, and quasi-isometry.

We classify surface Houghton groups, as well as their pure subgroups, up to isomorphism, commensurability, and quasi-isometry.

Paper Structure (7 sections, 8 theorems, 10 equations, 3 figures)

This paper contains 7 sections, 8 theorems, 10 equations, 3 figures.

Theorem 1.1

For $g,g' \geq 0$, $h,h'\geq 1$, and $r,r' \geq 2$, the groups $B(g,h,r)$ and $B(g',h',r')$:

Figure 1: An example demonstrating that $B(0,2,2)$ cannot be isomorphic to $B(1,2,2)$. On the top surface, $C$ is the core (denoted in blue) for $B(0,2,2)$. This coincides with a suited subsurface, $W$, for the involution $\rho \in B(0,2,2)$. On the bottom surface, $\rho'$ is a potential image of $\rho$ under an alleged isomorphism, $C'$ (in blue) is the core, and $W'$ (in green) is a suited subsurface for $\rho'$. Fixed points of the two maps are denoted in orange. Note that $\rho$ has exactly two fixed points and $\rho'$ must have at least four.
Figure 2: An example of a rigid structure $\mathcal{R}_{2,2,3}$ (right) that is engulfed by the rigid structure $\mathcal{R}_{0,1,3}$ (left). The two cores are shaded orange and the curves that are drawn are the boundaries of the pieces.
Figure 3: An example of the labeling in \ref{['lem:engulf']} for $\mathcal{R}_{0,1,r}$ engulfing $\mathcal{R}_{g,3,r}$. Note that the shift map to the right will induce the cyclic permutation $1 \mapsto 2 \mapsto 3 \mapsto 1$.