Critical points for the Hausdorff dimension of pairs of pants (Q1691971)

This paper mainly focuses on the dependence of the Hausdorff dimension of the limit set of a Fuchsian group on the geometry of the associated Riemannian surface. In particular, the group can be parametrized as \[ \underline{b}=(b_1, b_2, b_3) \in \Delta_b=\{(b_1, b_2, b_3)\in \mathbb{R}^3: b_1+b_2+b_3=b \}, \] where the boundary geodesic of the pant has lengths \(2b_1, 2b_2, 2b_3\). The authors consider the properties of the dimension map \(\Delta_b \to \mathbb{R}_+\) through the Selberg function \[ \mathbb{Z}_{\underline{b}}(s)=1+\sum_{n=1}^\infty a_{2n}(s,\underline{b}). \] The authors determine the critical points of the map and prove that the critical point \((b/3, b/3, b/3)\) is a local minimum if \(b\) is sufficiently large.