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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.
more » « less Award ID(s):
 2106906
 NSFPAR ID:
 10524897
 Publisher / Repository:
 Journal of the London Mathematical Society
 Date Published:
 Journal Name:
 Journal of the London Mathematical Society
 Volume:
 109
 Issue:
 6
 ISSN:
 00246107
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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