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Friedrich Martin Schneider

Publications and source records attributed to Friedrich Martin Schneider.

At least 19 recordsLinked to original sources
arXiv2026

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

We prove that the projective unit group $\mathrm{PGL}(R)$, i.e., the quotient of the unit group $\mathrm{GL}(R)$ modulo its center, of any non-discrete irreducible, continuous ring $R$ is simple. Moreover, we show that $\mathrm{GL}(R)$ 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 $\mathrm{GL}(R)$ on a metric space has bounded orbits. It follows that every action of $\mathrm{GL}(R)$ by isometries on a non-empty complete $\mathrm{CAT}(0)$ space admits a fixed point. In particular, $\mathrm{GL}(R)$ possesses Serre's properties $(FH)$ and $(FA)$. Furthermore, our results entail that $\mathrm{PGL}(R)$ 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 $\mathrm{GL}(R)$ is verbally simple.

math.GR↗
arXiv2026

Hyperlinearity via amenable near representations

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.

math.GR↗
arXiv2026

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

We prove that, if $R$ is a non-discrete irreducible, continuous ring, then its unit group $\mathrm{GL}(R)$, 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.

math.GR↗
arXiv2026

Whirliness beyond amenability

Given a non-atomic standard probability space $(Ω,μ)$, we exhibit examples of non-amenable closed topological subgroups of $\mathrm{Aut}(Ω,μ)$ 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.

math.FA↗
arXiv2026

Width bounds and Steinhaus property for unit groups of continuous rings

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

math.GR↗
arXiv2026

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

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.

math.GR↗
arXiv2025

Amenability and skew-amenability of actions of topological groups

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.

math.GR↗
arXiv2024

Groups without unitary representations, submeasures, and the escape property

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 $L^0(ϕ, G)$, 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 $L^0(ϕ,\mathbb{R})$ 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 $L^0(ϕ, G)$ to $L^0(μ, H)$, where $ϕ$ is pathological, $μ$ is a measure, $G$ is a topological group, and $H$ is a topological group with the escape property.

math.RT↗
arXiv2024

Entropy of group actions beyond uniform lattices

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.

math.DS↗
arXiv2024

Echeloned Spaces

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.

math.LO↗
arXiv2023

Group von Neumann algebras, inner amenability, and unit groups of continuous rings

We prove that, if a discrete group $G$ is not inner amenable, then the unit group of the ring of operators affiliated with the group von Neumann algebra of $G$ is non-amenable with respect to the topology generated by its rank metric. This provides examples of non-discrete irreducible, continuous rings (in von Neumann's sense) whose unit groups are non-amenable with regard to the rank topology. Our argument establishes and uses connections with Eymard--Greenleaf amenability of the action of the unitary group of a $\mathrm{II}_{1}$ factor on the associated space of projections of a fixed trace.

math.OA↗
arXiv2023

Concentration of invariant means and dynamics of chain stabilizers in continuous geometries

We prove a concentration inequality for invariant means on topological groups, namely for such adapted to a chain of amenable topological subgroups. The result is based on an application of Azuma's martingale inequality and provides a method for establishing extreme amenability. Building on this technique, we exhibit new examples of extremely amenable groups arising from von Neumann's continuous geometries. Along the way, we also answer a question by Pestov on dynamical concentration in direct products of amenable topological groups.

math.FA↗
arXiv2023

On invariant means and pre-syndetic subgroups

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.

math.GR↗
arXiv2023

Intrinsic Dimension for Large-Scale Geometric Learning

The concept of dimension is essential to grasp the complexity of data. A naive approach to determine the dimension of a dataset is based on the number of attributes. More sophisticated methods derive a notion of intrinsic dimension (ID) that employs more complex feature functions, e.g., distances between data points. Yet, many of these approaches are based on empirical observations, cannot cope with the geometric character of contemporary datasets, and do lack an axiomatic foundation. A different approach was proposed by V. Pestov, who links the intrinsic dimension axiomatically to the mathematical concentration of measure phenomenon. First methods to compute this and related notions for ID were computationally intractable for large-scale real-world datasets. In the present work, we derive a computationally feasible method for determining said axiomatic ID functions. Moreover, we demonstrate how the geometric properties of complex data are accounted for in our modeling. In particular, we propose a principle way to incorporate neighborhood information, as in graph data, into the ID. This allows for new insights into common graph learning procedures, which we illustrate by experiments on the Open Graph Benchmark.

cs.LG↗
arXiv2021

Skew-amenability of topological groups

We study skew-amenable topological groups, i.e., those admitting a left-invariant mean on the space of bounded real-valued functions left-uniformly continuous in the sense of Bourbaki. We prove characterizations of skew-amenability for topological groups of isometries and automorphisms, clarify the connection with extensive amenability of group actions, establish a Følner-type characterization, and discuss closure properties of the class of skew-amenable topological groups. Moreover, we isolate a dynamical sufficient condition for skew-amenability and provide several concrete variations of this criterion in the context of transformation groups. These results are then used to decide skew-amenability for a number of examples of topological groups built from or related to Thompson's group $F$ and Monod's group of piecewise projective homeomorphisms of the real line.

math.GR↗
arXiv2020

Intrinsic Dimension of Geometric Data Sets

The curse of dimensionality is a phenomenon frequently observed in machine learning (ML) and knowledge discovery (KD). There is a large body of literature investigating its origin and impact, using methods from mathematics as well as from computer science. Among the mathematical insights into data dimensionality, there is an intimate link between the dimension curse and the phenomenon of measure concentration, which makes the former accessible to methods of geometric analysis. The present work provides a comprehensive study of the intrinsic geometry of a data set, based on Gromov's metric measure geometry and Pestov's axiomatic approach to intrinsic dimension. In detail, we define a concept of geometric data set and introduce a metric as well as a partial order on the set of isomorphism classes of such data sets. Based on these objects, we propose and investigate an axiomatic approach to the intrinsic dimension of geometric data sets and establish a concrete dimension function with the desired properties. Our model for data sets and their intrinsic dimension is computationally feasible and, moreover, adaptable to specific ML/KD-algorithms, as illustrated by various experiments.

cs.AI↗