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Critical points for the Hausdorff dimension of pairs of pants. (English) Zbl 1386.37046

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.

MSC:

37F35 Conformal densities and Hausdorff dimension for holomorphic dynamical systems
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
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