preprints | Friedrich Martin Schneider

2024

Friedrich Martin Schneider, Sławomir Solecki

Groups without unitary representations, submeasures, and the escape property

Preprint (2024), 40 pages.

Abs arXiv

We give new examples of topological groups that do not have non-trivial continuous unitary representations, the so-called exotic groups. We prove that all groups of the form L⁰(ϕ,G), where ϕ is a pathological submeasure and G is a topological group, are exotic. This result extends, with a different proof, a theorem of Herer and Christensen on exoticness of L⁰(ϕ,ℝ) for ϕ pathological. It follows that every topological group embeds into an exotic one. In our arguments, we introduce the escape property, a geometric condition on a topological group, inspired by the solution to Hilbert’s fifth problem and satisfied by all locally compact groups, all non-archimedean groups, and all Banach–Lie groups. Our key result involving the escape property asserts triviality of all continuous homomorphisms from L⁰(ϕ,G) to L⁰(μ,H), where ϕ is pathological, μ is a measure, G is a topological group, and H is a topological group with the escape property.

2023

Maxime Gheysens, Bojana Pavlica, Christian Pech, Maja Pech, Friedrich Martin Schneider

Echeloned Spaces

Preprint (2023), 35 pages.

Abs arXiv

We introduce the notion of echeloned spaces - an order-theoretic abstraction of metric spaces. The first step is to characterize metrizable echeloned spaces. It turns out that morphisms between metrizable echeloned spaces are uniformly continuous or have a uniformly discrete image. In particular, every automorphism of a metrizable echeloned space is uniformly continuous, and for every metric space with midpoints the automorphisms of the induced echeloned space are precisely the dilations. Next we focus on finite echeloned spaces. They form a Fraisse class and we describe its Fraisse-limit both as the echeloned space induced by a certain homogeneous metric space and as the result of a random construction. Building on this we show that the class of finite ordered echeloned spaces is Ramsey. The proof of this result combines a combinatorial argument by Nesetril and Hubicka with a topological-dynamical point of view due to Kechris, Pestov, and Todorcevic. Finally, using the method of Katetov functors due to Kubis and Masulovic, we prove that the full symmetric group on a countable set topologically embeds into the automorphism group of the countable universal homogeneous echeloned space.
Vladimir Pestov, Friedrich Martin Schneider

On invariant means and pre-syndetic subgroups

To appear in Topology and its Applications, preprint (2023), 16 pages.

Abs arXiv

Beyond the locally compact case, equivalent notions of amenability diverge, and some properties no longer hold, for instance amenability is not inherited by topological subgroups. This investigation is guided by some amenability-type properties of groups of paths and loops. It is shown that a version of amenability called skew-amenability is inherited by pre-syndetic subgroups in the sense of Basso and Zucker (in particular, by co-compact subgroups). It follows that co-compact subgroups of amenable topological groups whose left and right uniformities coincide are amenable. We discuss a version of amenability belonging to P. Malliavin and M.-P. Malliavin: the existence of a mean on bounded Borel functions that is invariant under the left action of a dense subgroup. We observe that this property is in general strictly stronger than amenability, and establish for it Reiter- and Følner-type criteria. Finally, there is a review of open problems.

2022

Maksym Chaudkhari, Kate Juschenko, Friedrich Martin Schneider

Properties of group actions on orbits of an amenable equivalence relation and topological versions of Kesten’s theorem

Preprint (2022), 34 pages.

Abs arXiv

We obtain results connecting the uniform Liouville property of group actions on the classes of a countable Borel equivalence relation with amenability of this equivalence relation. We also discuss extensions of Kesten’s theorem to certain amenable topological groups and prove a version of this theorem for amenable groups with the SIN property. Furthermore, we discuss potential implications of generalizations of Kesten’s theorem for concentration inequalities on the inverted orbits of random walks on the classes of an amenable equivalence relation.
Till Hauser, Friedrich Martin Schneider

Entropy of group actions beyond uniform lattices

Preprint (2022), 31 pages.

Abs arXiv

Entropy of measure preserving or continuous actions of amenable discrete groups allows for various equivalent approaches. Among them are the ones given by the techniques developed by Ollagnier and Pinchon on the one hand and the Ornstein-Weiss lemma on the other. We extend these two approaches to the context of actions of amenable topological groups. In contrast to the discrete setting, our results reveal a remarkable difference between the two concepts of entropy in the realm of non-discrete groups: while the first quantity collapses to 0 in the non-discrete case, the second yields a well-behaved invariant for amenable unimodular groups. Concerning the latter, we moreover study the corresponding notion of topological pressure, prove a Goodwyn-type theorem, and establish the equivalence with the uniform lattice approach (for locally compact groups admitting a uniform lattice). Our study elaborates on a version of the Ornstein-Weiss lemma due to Gromov.