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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.
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Extensions of Veech groups I: A hyperbolic action
Abstract Given a lattice Veech group in the mapping class group of a closed surface , this paper investigates the geometry of , the associated ‐extension group. We prove that is the fundamental group of a bundle with a singular Euclidean‐by‐hyperbolic geometry. Our main result is that collapsing “obvious” product regions of the universal cover produces an action of on a hyperbolic space, retaining most of the geometry of . This action is a key ingredient in the sequel where we show that is hierarchically hyperbolic and quasi‐isometrically rigid.
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- Award ID(s):
- 2005368
- PAR ID:
- 10420793
- Publisher / Repository:
- Oxford University Press (OUP)
- Date Published:
- Journal Name:
- Journal of Topology
- Volume:
- 16
- Issue:
- 2
- ISSN:
- 1753-8416
- Page Range / eLocation ID:
- p. 757-805
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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