preprints

2026

1. Friedrich Martin Schneider

   On unit groups of purely infinite rings

   Preprint (2026), 55 pages.

   AbsarXiv

   We prove that the unit group of any purely infinite, regular, right self-injective ring admits a universal sequence of words (thus has strong distortion), is minimally almost periodic, has finite involution width, is verbally simple and verbally elliptic, and has uncountable uniform strong cofinality. These findings are consequences of more refined structural results, some of which provide concrete cardinal estimates involving the Goodearl–Boyle dimension of the ring. A central ingredient in our results is a comprehensive study of bounded normal generation, in particular with respect to certain classes of involutions. Along the way, we develop some abstract methods and concepts pertaining to geometric group theory.

2. Yannik Höll, Friedrich Martin Schneider

   Whirliness beyond amenability

   Preprint (2026), 25 pages.

   AbsarXiv

   Given a non-atomic standard probability space (Ω,μ), we exhibit examples of non-amenable closed topological subgroups of Aut(Ω,μ) whose natural near-action on (Ω,μ) is whirly. This answers a 2010 question by Pestov in the negative. The argument proceeds via constructing Gaussian near-actions from topologically faithful unitary representations of non-amenable whirly Polish groups.

2025

1. Friedrich Martin Schneider

   Projective simplicity and Bergman’s property for unit groups of continuous rings

   Preprint (2025), 24 pages.

   AbsarXiv

   We prove that the projective unit group PGL(R), i.e., the quotient of the unit group GL(R) modulo its center, of any non-discrete irreducible, continuous ring R is simple. Moreover, we show that GL(R) has uncountable strong cofinality, that is, it is not the union of a countable chain of proper subgroups and it has finite width with respect to any generating set. Equivalently, every isometric action of GL(R) on a metric space has bounded orbits. It follows that every action of GL(R) by isometries on a non-empty complete CAT(0) space admits a fixed point. In particular, GL(R) possesses Serre’s properties (FH) and (FA). Furthermore, our results entail that PGL(R) has bounded normal generation. In turn, we answer two questions by Carderi and Thom. Finally, building on work of Larsen and Shalev, we prove that GL(R) is verbally simple.

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

   To appear in Groups, Geometry, and Dynamics (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.

2022

1. Maksym Chaudkhari, Kate Juschenko, Friedrich Martin Schneider

   Amenable equivalence relations, Kesten’s property, and measurable lamplighters

   Preprint (2022), 37 pages.

   AbsarXiv

   We prove a characterization of the amenability of countable Borel equivalence relations in terms of the uniform Liouville property for group actions on their classes. Furthermore, inspired by a well-known amenability criterion for locally compact groups due to Kesten, we study return probabilities for random walks, and in particular a limiting condition that we call Kesten’s property, on general topological groups. We show that every amenable topological group with small invariant neighborhoods indeed has Kesten’s property. For measurable lamplighter groups associated with countable Borel equivalence relations, we establish a connection between Kesten’s property and anti-concentration inequalities for the inverted orbits of random walks on the equivalence classes. This allows us to construct an amenable contractible Polish group without Kesten’s property.