Abstract.
Consider the tessellation of the hyperbolic plane by m-gons, ℓ per vertex. In its 1-skeleton, we compute the growth series of vertices, geodesics, tuples of geodesics with common extremities. We also introduce and enumerate holly trees, a family of proper loops in these graphs. We then apply Grigorchuk’s result relating cogrowth and random walks to obtain lower estimates on the spectral radius of the Markov operator associated with a symmetric random walk on these graphs.
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Received 19 September 2001; in revised form 23 December 2001
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Bartholdi, L., Ceccherini-Silberstein, T. Growth Series and Random Walks on Some Hyperbolic Graphs. Mh Math 136, 181–202 (2002). https://doi.org/10.1007/s006050200043
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DOI: https://doi.org/10.1007/s006050200043