preprints

2024

1. Josefin Bernard, Friedrich Martin Schneider

   Width bounds and Steinhaus property for unit groups of continuous rings

   Preprint (2024), 61 pages.

   AbsarXiv

   We prove an algebraic decomposition theorem for the unit group GL(R) of an arbitrary non-discrete irreducible, continuous ring R (in von Neumann’s sense), which entails that every element of GL(R) is both a product of 7 commutators and a product of 16 involutions. Combining this with further insights into the geometry of involutions, we deduce that GL(R) has the so-called Steinhaus property with respect to the natural rank topology, thus every homomorphism from GL(R) to a separable topological group is necessarily continuous. Due to earlier work, this has further dynamical ramifications: for instance, for every action of GL(R) by homeomorphisms on a non-void metrizable compact space, every element of GL(R) admits a fixed point in the latter. In particular, our results answer two questions by Carderi and Thom, even in generalized form.

2. Vadim Alekseev, Hiroshi Ando, Friedrich Martin Schneider, Andreas Thom

   Amenability and skew-amenability of actions of topological groups

   Preprint (2024), 33 pages.

   AbsarXiv

   We define and study notions of amenability and skew-amenability of continuous actions of topological groups on compact topological spaces. Our main motivation is the question under what conditions amenability of a topological group passes to a closed subgroup. Other applications include the understanding of the universal minimal flow of various non-amenable groups.

2023

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

   Echeloned Spaces

   Preprint (2023), 35 pages.

   AbsarXiv

   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.

2022

1. 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), 38 pages.

   AbsarXiv

   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.