On the critical exponent of infinitely generated Veech groups

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.