On the Critical Exponent of Infinitely Generated Veech Groups

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 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. Hubert and Schmidt showed that these Veech groups are infinitely generated and of the first kind. We use a result of Roblin and Tapie which connects the critical exponent of the Veech group of the covering 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.