Higher rank confining subsets and hyperbolic actions of solvable groups

Abbott, Carolyn R.; Balasubramanya, Sahana; Rasmussen, Alexander J.

doi:10.1016/j.aim.2023.109045

Advances in Mathematics

2023

DOI: 10.1016/j.aim.2023.109045

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Higher rank confining subsets and hyperbolic actions of solvable groups

Carolyn R. Abbott¹,

Sahana Balasubramanya²,

Alexander J. Rasmussen³

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2024

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Valuations, completions, and hyperbolic actions of metabelian groups

Abbott,

Balasubramanya,

Rasmussen

2024

Journal of London Math Soc

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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.

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Valuations, completions, and hyperbolic actions of metabelian groups

Abbott,

Balasubramanya,

Rasmussen

2024

Journal of London Math Soc

Self Cite

View full text Add to dashboard Cite

show abstract

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Higher rank confining subsets and hyperbolic actions of solvable groups

Cited by 1 publication

References 13 publications

Valuations, completions, and hyperbolic actions of metabelian groups

Valuations, completions, and hyperbolic actions of metabelian groups

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