Automorphism groups of cyclic products of groups

TL;DR

This paper studies the automorphism groups of cyclic products of groups, revealing their structure and properties through geometric actions, including the extension of actions to Davis complexes and implications like Tits Alternative and hyperbolicity.

Contribution

It provides a comprehensive geometric analysis of automorphism groups of cyclic group products, including explicit computations and property characterizations.

Findings

Automorphism group action extends to Davis complex for cyclic products of at least five groups.

Complete computation of automorphism groups in this setting.

Demonstrates properties like Tits Alternative, acylindrical hyperbolicity, and absence of property (T).

Abstract

This article initiates a geometric study of the automorphism groups of general graph products of groups, and investigates the algebraic and geometric structure of automorphism groups of cyclic product of groups. For a cyclic product of at least five groups, we show that the action of the cyclic product on its Davis complex extends to an action of the whole automorphism group. This action allows us to completely compute the automorphism group and to derive several of its properties: Tits Alternative, acylindrical hyperbolicity, lack of property (T).