Gluing diagrams part 1: A constructive solution for the Higman-Thompson group isomorphism problem

TL;DR

This paper develops a combinatorial tool called gluing diagrams to construct and identify isomorphisms between Higman-Thompson groups via shift pseudogroups of directed graphs.

Contribution

It introduces gluing diagrams as a constructive method to determine isomorphisms between Higman-Thompson groups, confirming a conjecture by Higman.

Findings

Gluing diagrams can produce isomorphisms between shift pseudogroups.

A procedure is described to explicitly construct these isomorphisms.

The method confirms the conjectured isomorphisms between Higman-Thompson groups.

Abstract

This paper introduces gluing diagrams a combinatorial tool to construct homomorphisms between the shift pseudogroups of directed graphs and thus also their full groups of shifts. We will establish which of these diagrams produce isomorphisms. As an application, using the interpretation of Higman-Thompson groups as full groups of shifts of specific graphs, we will describe a procedure that constructs gluing diagrams that explicitly describe the isomorphisms between Higman-Thompson groups, conjectured by Higman and whose existence was proven by Pardo arXiv:1006.1759.