On fixing boundary points of transitive hyperbolic graphs
Agelos Georgakopoulos, Matthias Hamann
TL;DR
This paper proves that certain infinite, planar, hyperbolic graphs cannot have a boundary point with a transitive stabilizer, addressing a question in geometric group theory.
Contribution
It provides a partial answer to a question by Kaimanovich and Woess about the structure of hyperbolic graphs and their boundary stabilizers.
Findings
No 1-ended, planar, hyperbolic graph has a transitive boundary point stabilizer.
The result constrains possible symmetries of hyperbolic graphs at boundary points.
Addresses a specific open question in the theory of hyperbolic groups and graphs.
Abstract
We show that there is no 1-ended, planar, hyperbolic graph such that the stabilizer of one of its hyperbolic boundary points acts transitively on the vertices of the graph. This gives a partial answer to a question by Kaimanovich and Woess.
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Taxonomy
TopicsGeometric and Algebraic Topology · Geometric Analysis and Curvature Flows · Advanced Operator Algebra Research