Spectral and dynamical invariants of Hecke triangle groups via transfer operators

2025/09/22 by Fedosova, Ksenia
#11K55 #11M36 #37C30 #37D35 #FOS: Mathematics #Number Theory (math.NT) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2509.17936

Abstract

In this paper, we consider Hecke triangle groups Γ_w for w>2 and associated infinite-volume orbifolds Γ_w \backslash ℍ. We show that the Selberg zeta function Z_Γw(s) can be approximated for s ∈ ℂ ∖ (1)/(2)(1-2 ℕ₀) by determinants of finite-dimensional matrices with an explicitly computed error term that decays exponentially as the matrix size increases. As an application, we evaluate the Hausdorff dimensions of Hecke triangle groups with high precision, explicitly compute the values of the corresponding Ruelle zeta functions at zero, and obtain estimates on orders of trivial zeroes of the Selberg zeta function.

Citations

• On the functional equation of twisted Ruelle zeta function and Fried's conjecture
• Hecke triangle groups, transfer operators and Hausdorff dimension
• On Fried's conjecture for compact hyperbolic manifolds
• Fractal Weyl bounds and Hecke triangle groups
• Meromorphic continuation of Selberg zeta functions with twists having non-expanding cusp monodromy
• Symbolic dynamics, automorphic functions, and Selberg zeta functions\n with unitary representations
• Period functions for Hecke triangle groups, and the Selberg zeta function as a Fredholm determinant

Related

• Hecke triangle groups, transfer operators and Hausdorff dimension
• Period functions for Hecke triangle groups, and the Selberg zeta function as a Fredholm determinant
• Computation of Selberg zeta functions on Hecke triangle groups
• Fractal Weyl bounds and Hecke triangle groups
• Symbolic dynamics, automorphic functions, and Selberg zeta functions\n with unitary representations