Equivariant dissipation in non-Archimedean groups. (English) Zbl 1465.37005
Summary: We prove that, if a topological group \(G\) has an open subgroup of infinite index, then every net of tight Borel probability measures on \(G\) UEB-converging to invariance dissipates in \(G\) in the sense of Gromov. In particular, this solves a 2006 problem by V. Pestov [Dynamics of infinite-dimensional groups. The Ramsey-Dvoretzky-Milman phenomenon. Providence, RI: American Mathematical Society (AMS) (2006; Zbl 1123.37003)]: for every left-invariant (or right-invariant) metric \(d\) on the infinite symmetric group \(\mathrm{Sym}(\mathbb{N})\), compatible with the topology of pointwise convergence, the sequence of the finite symmetric groups \((\mathrm{Sym} (n), d \upharpoonright_{\mathrm{Sym}(n)}, \mu_{\mathrm{Sym}(n)})_{n \in \mathbb{N}}\) equipped with the restricted metrics and their normalized counting measures dissipates, thus fails to admit a subsequence being Cauchy with respect to Gromov’s observable distance.
MSC:
| 37A15 | General groups of measure-preserving transformations and dynamical systems |
| 22F05 | General theory of group and pseudogroup actions |
| 32B05 | Analytic algebras and generalizations, preparation theorems |
| 28A33 | Spaces of measures, convergence of measures |
Citations:
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