Stabilizers in Higman-Thompson groups

TL;DR

This paper studies the structure of stabilizers of finite sets of rational points in Cantor space within Higman-Thompson groups, revealing their algebraic properties and classifying them up to isomorphism.

Contribution

It establishes that these stabilizers are iterated ascending HNN extensions, their commutator subgroups are simple, and provides a classification up to isomorphism.

Findings

Pointwise stabilizers are iterated ascending HNN extensions of $V_{n,q}$.

The commutator subgroup of the stabilizer is simple.

The abelianization of the stabilizer is computed.

Abstract

We investigate stabilizers of finite sets of rational points in Cantor space for the Higman-Thompson groups $V_{n,r}$. We prove that the pointwise stabilizer is an iterated ascending HNN extension of $V_{n,q}$ for any $q\geq 1$. We also prove that the commutator subgroup of the pointwise stabilizer is simple, and we compute the abelianization. Finally, for each $n$ we classify such pointwise stabilizers up to isomorphism.