Valuations, completions, and hyperbolic actions of metabelian groups

TL;DR

This paper classifies cobounded hyperbolic actions of many metabelian groups by linking hyperbolic geometry with commutative algebra, revealing their structure as actions on trees or Heintze groups.

Contribution

It introduces a novel algebraic framework to classify hyperbolic actions of metabelian groups, connecting geometric actions with ideal classification in ring completions.

Findings

Classified hyperbolic actions of abelian-by-cyclic groups

Linked geometric actions to ideal classification in algebraic structures

Identified actions as on trees or Heintze groups

Abstract

Actions on hyperbolic metric spaces are an important tool for studying groups, and so it is natural, but difficult, to attempt to classify all such actions of a fixed group. In this paper, we build strong connections between hyperbolic geometry and commutative algebra in order to classify the cobounded hyperbolic actions of numerous metabelian groups up to a coarse equivalence. In particular, we turn this classification problem into the problems of classifying ideals in the completions of certain rings and calculating invariant subspaces of matrices. We use this framework to classify the cobounded hyperbolic actions of many abelian-by-cyclic groups associated to expanding integer matrices. Each such action is equivalent to an action on a tree or on a Heintze group (a classically studied class of negatively curved Lie groups). Our investigations incorporate number systems, factorization in formal power series rings, completions, and valuations.