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An ergodic theorem for the quasi-regular representation of the free group
Group Theory · Submitted 4 Jan 2016
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We prove all the analog results in the case of the free group $\mathbb{F}_r$ of rank $r$ even if $\mathbb{F}_r$ is not the fundamental group of a closed manifold, and may have an arithmetic spectrum.
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The abstract, sentence by sentence
In \cite{BAMU}, an ergodic theorem \`a la Birkhoff-von Neumann for the action of the fundamental group of a compact negatively curved manifold on the boundary of its universal cover is proved. A quick corollary is the irreducibility of the associated unitary representation. These results are generalized \cite{BOYER} to the context of convex cocompact groups of isometries of a CAT(-1) space, using Theorem 4.1.1 of \cite{ROBLI}, with the hypothesis of non arithmeticity of the spectrum. We prove all the analog results in the case of the free group $\mathbb{F}_r$ of rank $r$ even if $\mathbb{F}_r$ is not the fundamental group of a closed manifold, and may have an arithmetic spectrum.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In \cite{BAMU}, an ergodic theorem \`a la Birkhoff-von Neumann for the action of the fundamental group of a compact negatively curved manifold on the boundary of its universal cover is proved. | background 1.00 | 0.04 | 0.05 | asserted 1.00 |
| s1 | A quick corollary is the irreducibility of the associated unitary representation. | result 0.55 | 0.15 | 0.06 | asserted 1.00 |
| s2 | These results are generalized \cite{BOYER} to the context of convex cocompact groups of isometries of a CAT(-1) space, using Theorem 4.1.1 of \cite{ROBLI}, with the hypothesis of non arithmeticity of the spectrum. | background 0.99 | 0.09 | 0.06 | asserted 1.00 |
| s3 | We prove all the analog results in the case of the free group $\mathbb{F}_r$ of rank $r$ even if $\mathbb{F}_r$ is not the fundamental group of a closed manifold, and may have an arithmetic spectrum. | contribution 0.90 | 0.96 | 0.92 | asserted 0.99 |
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Original abstract
In \cite{BAMU}, an ergodic theorem \`a la Birkhoff-von Neumann for the action of the fundamental group of a compact negatively curved manifold on the boundary of its universal cover is proved. A quick corollary is the irreducibility of the associated unitary representation. These results are generalized \cite{BOYER} to the context of convex cocompact groups of isometries of a CAT(-1) space, using Theorem 4.1.1 of \cite{ROBLI}, with the hypothesis of non arithmeticity of the spectrum. We prove all the analog results in the case of the free group $\mathbb{F}_r$ of rank $r$ even if $\mathbb{F}_r$ is not the fundamental group of a closed manifold, and may have an arithmetic spectrum.
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Uses the method or tool of 2 papers
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We prove it following the ideas of [ref] and [ref] replacing [ref] by Theorem [ref].
Equidistribution, ergodicity and irreducibility in CAT(-1) spaces · Introduction · 0.96
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Then, the classical argument of [ref] concludes the proof.
Boundary representations of hyperbolic groups · Results · 0.89
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