Semiregular Polytopes and Amalgamated C-groups

TL;DR

This paper explores the structure and construction of semiregular abstract polytopes, focusing on automorphism groups, amalgamation techniques, and finite examples derived from reflection groups over finite fields.

Contribution

It introduces a method to construct universal semiregular polytopes via group amalgamation and applies modular reduction to generate finite examples from reflection groups.

Findings

Existence of universal semiregular polytopes from compatible facets

Construction of finite semiregular polytopes using modular reduction

Analysis of automorphism groups with alternating regular facets

Abstract

In the classical setting, a convex polytope is said to be semiregular if its facets are regular and its symmetry group is transitive on vertices. This paper studies semiregular abstract polytopes, which have abstract regular facets, still with combinatorial automorphism group transitive on vertices. We analyze the structure of the automorphism group, focusing in particular on polytopes with two kinds of regular facets occurring in an "alternating" fashion. In particular we use group amalgamations to prove that given two compatible n-polytopes P and Q, there exists a universal abstract semiregular (n+1)-polytope which is obtained by "freely" assembling alternate copies of P and Q. We also employ modular reduction techniques to construct finite semiregular polytopes from reflection groups over finite fields.