Counting subgraphs in fftp graphs with symmetry

TL;DR

This paper proves the rationality of generating functions counting subgraph embeddings in certain symmetric graphs, extending classical results and applying to hyperbolic groups with quasi-convex subgroups.

Contribution

It generalizes previous work by establishing rational growth functions for convex subgraphs in graphs with the fftp, including Schreier coset graphs of hyperbolic groups.

Findings

Rationality of growth functions for convex subgraphs in fftp graphs

Falsification by fellow traveler property holds for Schreier coset graphs of hyperbolic groups

Existence of a uniform lower bound for growth rates regardless of generating set

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.