Víctor Hugo Yañez

Algebraic structure of locally finite groups with the Lévy property.

All topological spaces considered in this talk shall be assumed Hausdorff. A topological group G is said to be extremely amenable if each continuous action of G on a compact space X admits a fixed point. Each extremely amenable group is minimally almost periodic (\mathrm{MinAP}) in the sense of von Neumann, i.e., they admit no non-trivial continuous homomorphism to a compact group. A topological group G is said to have the Lévy property if it admits an increasing sequence \mathcal{G} = \{K_i : i \in \mathbb{N}\} of compact subgroups (equipped with their normalized Haar measures \mu_i) with the following properties:

  1. The union \bigcup \mathcal{G} is dense in G, and
  2. For each open neighborhood V of the identity of G and each sequence \{A_i : i \in \mathbb{N}\} of Borel subsets A_i \subseteq K_i satisfying \liminf_{i \to \infty} \mu_i(A_i) > 0, we have

        \[\lim_{i \to \infty} \mu_i(V \cdot A_i) = 1.\]

We say that the topological group G satisfies the strong Lévy property if each group K_i above can be taken finite. While each group with the Lévy property is extremely amenable, the converse readily fails: no Polish non-archimedean extremely amenable group can satisfy the Lévy property. Pestov proved in 2007 that the isometry group \mathrm{Iso}(\mathbb{U}) of the universal Urysohn metric space \mathbb{U} satisfies the strong Lévy property. Consequently, the union \bigcup \mathcal{G} of the witnessing sequence is a countable, locally finite (i.e., any of its finitely generated subgroups is finite) and dense subgroup of \mathrm{Iso}(\mathbb{U}) with the (strong) Lévy property. In this talk, we present new results on the algebraic structure of locally finite groups which admit a group topology with the Lévy property.

Results in this talk are from a joint research project with Wei Dai and Su Gao (Nankai University).