The infinite cyclohedron and its automorphism group

TL;DR

This paper introduces an infinite-dimensional analogue of cyclohedra, revealing its automorphism group as a semidirect product involving Thompson's group T, expanding understanding of symmetries in infinite polytopal complexes.

Contribution

It defines an infinite-dimensional cyclohedron and characterizes its automorphism group as a specific semidirect product involving Thompson's group T.

Findings

Automorphism group is a semidirect product with Thompson's group T.

Provides a new infinite-dimensional analogue of cyclohedra.

Connects symmetries of infinite polytopes to well-known infinite groups.

Abstract

Cyclohedra are a well-known infinite familiy of finite-dimensional polytopes that can be constructed from centrally symmetric triangulations of even-sided polygons. In this article we introduce an infinite-dimensional analogue and prove that the group of symmetries of our construction is a semidirect product of a degree 2 central extension of Thompson's infinite finitely presented simple group T with the cyclic group of order 2. These results are inspired by a similar recent analysis by the first author of the automorphism group of an infinite-dimensional associahedron.