preprints | Friedrich Martin Schneider

2025

Friedrich Martin Schneider

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

Preprint (2025), 21 pages.

Abs arXiv

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.
Friedrich Martin Schneider

Geometric properties of unit groups of von Neumann’s continuous rings

Preprint (2025), 36 pages.

Abs arXiv

We prove that, if R is a non-discrete irreducible, continuous ring, then its unit group GL(R), equipped with the topology generated by the rank metric, is topologically simple modulo its center, path-connected, locally path-connected, bounded in the sense of Bourbaki, and not admitting any non-zero escape function. All these topological insights are consequences of more refined geometric results concerning the rank metric, in particular with regard to the set of algebraic elements. Thanks to the phenomenon of automatic continuity, our results also have non-trivial ramifications for the underlying abstract groups.
Paula Kahl, Friedrich Martin Schneider

Hyperlinearity via amenable near representations

To appear in Transactions of the American Mathematical Society (2025), 51 pages.

Abs doi arXiv

We note a characterization of the amenability of unitary representations (in the sense of Bekka) via the existence of an orthonormal basis supporting an invariant probability charge. Based on this, we explore several natural notions of amenable near representations on Hilbert spaces. Establishing an analogue of a theorem about sofic groups by Elek and Szabó, we prove that a group is hyperlinear if and only if it admits an essentially free amenable near representation. This answers a question raised by Pestov and Kwiatkowska. For comparison, we also provide characterizations of Kirchberg’s factorization property, as well as Haagerup’s property, along similar lines.

2024

Josefin Bernard, Friedrich Martin Schneider

Width bounds and Steinhaus property for unit groups of continuous rings

Preprint (2024), 61 pages.

Abs arXiv

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.
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.

Abs arXiv

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

Maksym Chaudkhari, Kate Juschenko, Friedrich Martin Schneider

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

Preprint (2022), 37 pages.

Abs arXiv

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.