# Critical exponents of normal subgroups, the spectrum of group extended transfer operators, and Kazhdan distance

## TL;DR

This paper links the growth rate of normal subgroup orbits in Hadamard manifolds to Kazhdan distances via spectral analysis of transfer operators, revealing new connections between geometric group theory and representation theory.

## Contribution

It establishes a novel relationship between critical exponents of normal subgroups and Kazhdan distances using spectral analysis of transfer operators.

## Key findings

- Critical exponents mirror Kazhdan distances for quotient groups.
- Spectrum analysis of transfer operators provides new insights.
- Results apply to groups acting on pinched Hadamard manifolds.

## Abstract

For a pinched Hadamard manifold $X$ and a discrete group of isometries $\Gamma$ of $X$, the critical exponent $\delta_\Gamma$ is the exponential growth rate of the orbit of a point in $X$ under the action of $\Gamma$. We show that the critical exponent for any family $\mathcal{N}$ of normal subgroups of $\Gamma_0$ has the same coarse behaviour as the Kazhdan distances for the right regular representations of the quotients $\Gamma_0/\Gamma$. The key tool is to analyse the spectrum of transfer operators associated to subshifts of finite type, for which we obtain a result of independent interest.