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On the critical exponent of infinitely generated Veech groups

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Abstract

We prove the existence of Veech groups having a critical exponent strictly greater than any elementary Fuchsian group (i.e. \({>}\frac{1}{2}\)) but strictly smaller than any lattice (i.e. \({<}1\)). More precisely, every affine covering Y of a primitive L-shaped Veech surface X ramified over the singularity and a non-periodic connection point \(P\in X\) has such a Veech group \({{\mathrm{SL}}}(Y)\). Hubert and Schmidt (Duke Math J 123, 49–69 2004) showed that these Veech groups are infinitely generated and of the first kind. We use a result of Roblin and Tapie (Monogr Enseign Math 43:61–92, 2013) which connects the critical exponent of \({{\mathrm{SL}}}(Y)\) with the Cheeger constant of the Schreier graph of \({{\mathrm{SL}}}(X)/{{\mathrm{Stab}}}_{{{\mathrm{SL}}}(X)}(P)\). The main task is to show that the Cheeger constant is strictly positive, i.e. the graph is non-amenable. In this context, we introduce a measure of the complexity of connection points that helps to simplify the graph to a forest for which non-amenability can be seen easily.

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Notes

  1. Note that \(\mathbb {H}=\mathbb {H}^2=\mathbb {H}^{1+1}\).

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Acknowledgments

This paper summarizes and augments the results of the author’s dissertation of the same title. He thanks the European Research Council ERC-StG 257137 for financial support as well as J. Cuno, P. Hubert, M. Möller, S. Tapie and J. Zachhuber for useful discussions.

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  1. Darmstadt, Germany

    Ralf Lehnert

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  1. Ralf Lehnert
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Lehnert, R. On the critical exponent of infinitely generated Veech groups. Math. Ann. 368, 1017–1058 (2017). https://doi.org/10.1007/s00208-016-1462-6

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