
> 
> Let $G$ be a finitely generated Fuchsian group.
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(i.e. a discrete subgroup of $\mathrm{PSL}\_2(\mathbb{R})$).



> 
> Is it true that $d(G) < 2\beta\_{2}^1(G) + 1$ ?
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Here, $\beta\_{2}^1(G)$ stands for the first $L^2$-Betti number of $G$, and $d(G)$ is the smallest cardinality of a generating set of $G$.



An inequality for Fuchsian groups?
A priori the inequality seems unlikely because for a group containing an index k surface subgroup the L^2-Betti numbers are those of the surface group divided by k, so the right hand side gets approximately divided by k, but one wouldn't expect the same for the left hand side.


To get an explicit counterexample look at the action of a Hurwitz group on a genus g surface, the Fuchsian group corresponding to the quotient orbifold being the (2,3,7)-triangle group.


The Hurwitz group has order 84(g-1) and the first L^2-Betti number of the closed surface is (2g-2), so for the (2,3,7)-triangle group it is 1/42. So the right hand side is 1+1/21. But the (2,3,7)-triangle group is certainly not cyclic, so $d(G)= 2$.


