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Counting subgraphs in fftp graphs with symmetry

Part of: Special aspects of infinite or finite groups Combinatorics

Published online by Cambridge University Press:  27 November 2019

YAGO ANTOLÍN
YAGO ANTOLÍN*
Affiliation:
Departamento de Matemáticas, Facultad de Ciencas, Universidad Autónoma de Madrid Cantoblanco, Ciadad Universitaria, 28049 Madrid, Spain Instituto de Ciencias Matemáticas, Madrid, Spain. e-mail: [email protected]
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Abstract

Following ideas that go back to Cannon, we show the rationality of various generating functions of growth sequences counting embeddings of convex subgraphs in locally-finite, vertex-transitive graphs with the (relative) falsification by fellow traveler property (fftp). In particular, we recover results of Cannon, of Epstein, Iano–Fletcher and Zwick, and of Calegari and Fujiwara. One of our applications concerns Schreier coset graphs of hyperbolic groups relative to quasi-convex subgroups, we show that these graphs have rational growth, the falsification by fellow traveler property, and the existence of a lower bound for the growth rate independent of the finite generating set and the infinite index quasi-convex subgroup.

MSC classification

Primary: 20F67: Hyperbolic groups and nonpositively curved groups
Secondary: 05A15: Exact enumeration problems, generating functions
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 170 , Issue 2 , March 2021 , pp. 327 - 353
Copyright
© Cambridge Philosophical Society 2019

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