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arXiv · 1702.06115

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

Rhiannon Dougall
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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.

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Rhiannon Dougall. 2017-02-20. Critical exponents of normal subgroups, the spectrum of group extended transfer operators, and Kazhdan distance. https://arxiv.org/abs/1702.06115

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