arXiv 1702.061152017-02-20math.DSmath.DG

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

Rhiannon Dougall

备注：: 24 pages

英文摘要：

For a pinched Hadamard manifold $X$ and a discrete group of isometries $Γ$ of $X$, the critical exponent $δ_Γ$ is the exponential growth rate of the orbit of a point in $X$ under the action of $Γ$. We show that the critical exponent for any family $\mathcal{N}$ of normal subgroups of $Γ_0$ has the same coarse behaviour as the Kazhdan distances for the right regular representations of the quotients $Γ_0/Γ$. 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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