Regular and semi-regular representations of groups by posets

TL;DR

This paper explores conditions under which groups, including infinite and certain complex groups, can be represented as automorphism groups of posets with specific orbit structures, extending known results to broader classes.

Contribution

It establishes new criteria for groups to have regular or semi-regular poset representations with two orbits, especially relating to Cayley graphs and extensions of groups.

Findings

Groups with Cayley graphs locally like free groups have semi-regular two-orbit representations.

Extensions of integers by such groups admit regular representations.

Applications include finite simple, hyperbolic, random, and indicable groups.

Abstract

By a result of Babai, with finitely many exceptions, every group $G$ admits a semi-regular poset representation with three orbits, that is, a poset $P$ with automorphism group $\textrm{Aut}(P) \simeq G$ such that the action of $\textrm{Aut}(P)$ on the underlying set is free and with three orbits. Among finite groups, only the trivial group and $\mathbb{Z}_2$ have a regular poset representation (i.e. semi-regular with one orbit), however many infinite groups admit such a representation. In this paper we study non-necessarily finite groups which have a regular representation or a semi-regular representation with two orbits. We prove that if $G$ admits a Cayley graph which is locally the Cayley graph of a free group, then it has a semi-regular representation of height 1 with two orbits. In this case we will see that any extension of the integers by $G$ admits a regular representation. Applications are given to finite simple groups, hyperbolic groups, random groups and indicable groups.