Actions on trees, automorphism groups, free products and property FA

Lead Research Organisation: University of Southampton
Department Name: Sch of Mathematical Sciences

Abstract

This is a project in the area of geometric group theory, which studies the algebraic objects of groups via their actions on geometric objects. In particular, this project will be concerned with the algebraic information that can be gleaned from actions of discrete groups on trees.

The aims are to combine the strongly geometric language of translation length functions, along with the more combinatorial descriptions of Bass to understand actions and their properties.

The preliminary objectives will be to furnish a new proof that the LERF property passes to free products and to study the automorphism groups of free products and characterise when these groups have property FA. This is timely research as there has been a great deal of recent interest in the much stronger property T for automorphism groups of free groups.

Publications

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Andrew N (2021) Serre's Property (FA) for automorphism groups of free products in Journal of Group Theory

Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509747/1 30/09/2016 29/09/2021
1949273 Studentship EP/N509747/1 27/09/2017 30/03/2021 Naomi Andrew
 
Description The work involved investigating several aspects of group theory connected to actions on trees. This involved connecting these actions to properties of subgroups (known as "separability"), and to the automorphisms of certain classes of groups with actions on trees (free products and free-by-cyclic groups). In the case of free products, it is shown (under some natural conditions - for instance a free product of finite groups) when the automorphism group itself admits actions on trees, which depends largely on the number of times each isomorphism class appears in the decomposition as a free product.

For free-by-cyclic groups, and their automorphisms, the aim was to build actions on trees allowing study of the automorphisms. This involved introducing "nearly canonical trees", which have actions respected by a finite index subgroup of (outer) automorphisms, and using a theorem of Bass and Jiang to understand that subgroup. The conclusion was that in low rank (up to F3-by-cyclic) or "growth" (up to linearly growing defining automorphism) cases, the (outer) automorphism groups are finitely generated.

This has resulted in three papers, one already published and two in the revision process.
Exploitation Route The outcomes contribute to a growing body of knowledge in geometric group theory, and interact with problems involving Out(Fn) (such as the conjugacy problem). In particular the idea of nearly canonical trees shows promise to applications to new problems.
Sectors Other