arXiv subjects

Christian Rosendal

Publications and source records attributed to Christian Rosendal.

At least 19 recordsLinked to original sources

arXiv2026

Transfinite Schreier families and cardinal invariants

We study the dependence of the transfinite Schreier hierarchy on the choice of fundamental sequences for the countable limit ordinals. With each Schreier family we associate an interval endpoint function. We prove that the bounding number is equal to $ω_1$ exactly when the fundamental sequences may be chosen so that these endpoint functions form an unbounded family in the eventual domination order. Equivalently, the hierarchy then satisfies the tail-covering property isolated by Shiliaev, or every infinite compact interval family has infinite intersection with some member of the hierarchy. We also prove that the dominating number is equal to $ω_1$ exactly when the fundamental sequences may be chosen so that every compact family of finite subsets of $\{2,3,\ldots\}$ is contained in one Schreier family. The latter equivalence combines Fremlin's cofinality theorem for compact subsets of the rationals with an absorption construction for compact families.

math.FA↗

arXiv2026

Set theory, logic, and homeomorphism groups of manifolds

We investigate the relationship between axiomatic set theory and the first-order theory of homeomorphism groups of manifolds in the language of group theory, concentrating on first-order rigidity and type versus conjugacy. We prove that under the axiom of constructibility (i.e.~{V=L}), homeomorphism groups of arbitrary connected manifolds are first-order rigid, and that the conjugacy class of a homeomorphism of a manifold is determined by its type. In contradistinction, under the regularity hypothesis that every projective set of reals has the Baire property, we show that in all dimensions greater than one there exist pairs of noncompact, connected manifolds whose homeomorphism groups are elementarily equivalent but which are not homeomorphic. We also show, under the same Baire-property hypothesis, that every manifold of positive dimension admits pairs of homeomorphisms with the same type which are not conjugate to each other. Projective determinacy implies the Baire-property hypothesis, so the corresponding consequences under PD follow immediately. Finally, we show that infinitary formulas do determine conjugacy classes of homeomorphisms and homeomorphism types of manifolds; specifically, the conjugacy class of a homeomorphism of an arbitrary manifold is determined by a single $L_{ω_1ω}$ formula. Similarly, the homeomorphism type of an arbitrary connected manifold is determined by a single $L_{ω_1ω}$ sentence.

math.GT↗

arXiv2026

The geometrisation problem for topological groups

This paper presents a framework for assigning intrinsic geometric structures to topological groups using only the data provided by their topological and algebraic structure. The geometrisation spits into small-scale and large-scale components, formalised respectively through local Lipschitz and quasimetric categories that, in turn, are definable from the canonical left uniform and left coarse structures of the group. For Polish groups, the paper characterises metrisability of the left coarse structure in terms of local boundedness, countable coverings by bounded sets, and the existence of compatible coarsely proper left-invariant metrics. It then introduces minimal metrics, which determine local Lipschitz structure, and maximal metrics, which determine quasimetric structure, and provides intrinsic characterisations of both. When both structures exist, they combine into a single canonical Lipschitz structure. Our framework is subsequently applied to specific examples such as homeomorphism groups, non-Archimedean Polish groups and automorphism groups of Fraïssé limits.

math.GR↗

arXiv2026

Coarse Structures on Homogeneous Spaces

Given a closed normal subgroup $H$ of a topological group $G$, we address the question of whether the left coarse structure on the quotient group $G/H$ equals the quotient of the left coarse structure on $G$. We provide a counterexample among Polish groups, namely, the mapping class group of the Loch Ness monster surface seen as a quotient of the mapping class group of the punctured Loch Ness monster surface, and establish both equivalent and sufficient conditions for when this holds in special settings. The latter are formulated in terms of liftings of bounded sets, existence of transversals and metrisability of the left coarse structure of $G$ restricted to $H$.

math.GR↗

arXiv2026

Coordinate systems in Banach spaces and lattices

Using methods of descriptive set theory, in particular, the determinacy of infinite games of perfect information, we answer several questions from the literature regarding different notions of bases in Banach spaces and lattices. For the case of Banach lattices, our results follow from a general theorem stating that (under the assumption of analytic determinacy), every $σ$-order basis $(e_n)$ for a Banach lattice $X=[e_n]$ is a uniform basis, and every uniform basis is Schauder. Moreover, the notions of order and $σ$-order bases coincide when $X=[e_n].$ Regarding Banach spaces, we address two problems concerning filter Schauder bases for Banach spaces, i.e., in which the norm convergence of partial sums is replaced by norm convergence along some appropriate filter on $\mathbb N$. We first provide an example of a Banach space admitting such a filter Schauder basis, but no ordinary Schauder basis. Secondly, we show that every filter Schauder basis with respect to an analytic filter is also a filter Schauder basis with respect to a Borel filter.

math.FA↗

arXiv2026

A Classification of Order Convergence via a Transfinite Fatou Hierarchy

We investigate the descriptive complexity of order convergence in separable Banach lattices. While uniform convergence is Borel and $σ$-order convergence is known to be ${\bf Δ}^1_2$, it is unclear in general when $σ$-order convergence is analytic. We introduce a transfinite hierarchy of weakenings of the classical Fatou property, indexed by countable ordinals, and show that it provides a complete structural classification of this definability problem. For a separable Banach lattice $X$, we prove that the following are equivalent: (i) the set of decreasing positive sequences with infimum zero is Borel; (ii) $σ$-order convergence is analytic; and (iii) $X$ satisfies the $α$-Fatou property for some countable ordinal $α$. We further establish that the hierarchy is proper: for every countable ordinal $α$ there exists a separable Banach lattice with a countable $π$-basis that fails to be $α$-Fatou, but is $β$-Fatou for some $β>α$. Thus the Borel definability of order convergence is governed by a canonical ordinal invariant intrinsic to the lattice, and the descriptive complexity can be arbitrarily high below $ω_1$. These results identify projective complexity as a genuine structural invariant in Banach lattice theory.

math.FA↗

arXiv2025

Aspects of automatic continuity

A general overview of the phenomenon of automatic continuity of homomorphisms between Polish groups is given. In particular, we study variants and improvements of the closed graph theorem, applying these to the problem of continuity of universally measurable homomorphisms and also to gauge the amount of choice needed to construct discontinuous homomorphisms between Polish groups. Furthermore, we provide a simple proof of automatic continuity in the context of homeomorphism groups of compact manifolds and a complete reworking of automatic continuity theory in the context of isometry groups of highly homogeneous complete metric structures.

math.GR↗

arXiv2025

Amenability, Optimal Transport and Abstract Ergodic Theorems

Using tools from the theory of optimal transport, we establish several results concerning isometric actions of amenable topological groups with potentially unbounded orbits. Specifically, suppose $d$ is a compatible left-invariant metric on an amenable topological group $G$ with no non-trivial homomorphisms to $\mathbb R$. Then, for every finite subset $E\subseteq G$ and $ε>0$, there is a finitely supported probability measure $β$ on $G$ such that $$ \max_{g,h\in E}\, {\sf W}(βg, βh)<ε, $$ where ${\sf W}$ denotes the Wasserstein distance between probability measures on the metric space $(G,d)$. When $d$ is the word metric on a finitely generated group $G$, this strengthens a well known theorem of Reiter and, when $d$ is bounded, recovers a result of Schneider and Thom. Furthermore, when $G$ is locally compact, $β$ may be replaced by an appropriate probability density $f\in L^1(G)$. Also, when $G\curvearrowright X$ is a continuous isometric action on a metric space, the space of Lipschitz functions on the quotient $X/\!\!/G$ is isometrically isomorphic to a $1$-complemented subspace of the Lipschitz functions on $X$. And, when additionally $G$ is skew-amenable, there is a $G$-invariant contraction $$ \mathfrak {Lip}\, X \overset S\longrightarrow\mathfrak{Lip}(X/\!\!/G) $$ so that $(Sϕ\big)\big(\overline{Gx}\big)=ϕ(x)$ whenever $ϕ$ is constant on every orbit of $G\curvearrowright X$. This latter extends results of Cuth and Doucha from the setting of locally compact or balanced groups.

math.FA↗

arXiv2025

Continuity of measurable cocycles

Suppose $G\curvearrowright X$ is a Polish group action, $H$ is a Polish group and $G\times X\oversetψ\longrightarrow H$ is a cocycle that is continuous in the second variable. If $ψ$ is either Baire measurable or is $λ\times μ$-measurable with respect to a Haar measure $λ$ on $G$ and a fully supported $σ$-finite Borel measure $μ$ on $X$, then $ψ$ is jointly continuous.

math.GR↗

arXiv2024

Asymptotically spherical groups

We define a notion of asymptotically spherical topological groups, which says that spheres of large radius with respect to any maximal length function are still spherical with respect to any other maximal length function. This is a strengthening of a related condition introduced by Sebastian Hurtado, which we call bounded eccentricity. Our main result is a partial characterization of which groups are asymptotically spherical, and we also give an example of a discrete, bounded eccentric group who fails to be asymptotically spherical.

math.GR↗

arXiv2023

Abstract embeddability ranks

We describe several ordinal indices that are capable of detecting, according to various metric notions of faithfulness, the embeddability between pairs of Polish spaces. These embeddability ranks are of theoretical interest but seem difficult to estimate in practice. Embeddability ranks, which are easier to estimate in practice, are embeddability ranks generated by Schauder bases. These embeddability are inspired by the nonlinear indices à la Bourgain from \cite{BLMS_FM}. In particular, we resolve a problem \cite[Problem 3.10]{BLMS_FM} regarding the necessity of additional set-theoretic axioms regarding the main coarse universality result of \cite{BLMS_FM}.

math.MG↗

arXiv2023

Separation ratios of maps between Banach spaces

Under the weak assumption on a Banach space $E$ that $E\oplus E$ embeds isomorphically into $E$, we provide a characterisation of when a Banach space $X$ coarsely embeds into $E$ via a single numerical invariant.

math.FA↗

arXiv2022

On uniform and coarse rigidity of $L^p([0,1])$

If $X$ is an almost transitive Banach space with amenable isometry group (for example, if $X=L^p([0,1])$ with $1\leqslant p<\infty$) and $X$ admits a uniformly continuous map $X\oversetϕ\longrightarrow E$ into a Banach space $E$ satisfying $$\inf_{\|x-y\|=r} \| ϕ(x)-ϕ(y)\|>0 $$ for some $r>0$, then $X$ admits a simultaneously uniform and coarse embedding into a Banach space $V$ that is finitely representable in $L^2(E)$.

math.FA↗

arXiv2022

Finite conjugacy classes and split exact cochain complexes

We study the cohomology of isometric group actions on (super) reflexive Banach spaces with a focus on the relation between finite conjugacy classes and split exactness of cochain complexes. In particular, we show that, if a uniformly convex Banach module has no almost invariant vectors under the FC-centre of the acting group, then the associated cochain complex is split exact. Other similar rigidity results are established that are related to prior work of Bader - Furman - Gelander - Monod, Bader - Rosendal - Sauer and Nowak.

math.GR↗

arXiv2018

Light groups of isomorphisms of Banach spaces and invariant LUR renormings

Megrelishvili defines \emph{light groups} of isomorphisms of a Banach space as the groups on which the Weak and Strong Operator Topologies coincide, and proves that every bounded group of isomorphisms of Banach spaces with the Point of Continuity Property (PCP) is light. We investigate this concept for isomorphism groups $G$ of classical Banach spaces $X$ without the PCP, specially isometry groups, and relate it to the existence of $G$-invariant LUR or strictly convex renormings of $X$.

math.FA↗

arXiv2016

Lipschitz structure and minimal metrics on topological groups

We discuss the problem of deciding when a metrisable topological group $G$ has a canonically defined local Lipschitz geometry. This naturally leads to the concept of minimal metrics on $G$, that we characterise intrinsically in terms of a linear growth condition on powers of group elements. Combining this with work on the large scale geometry of topological groups, we also identify the class of metrisable groups admitting a canonical global Lipschitz geometry. In turn, minimal metrics connect with Hilbert's fifth problem for completely metrisable groups and we show, assuming that the set of squares is sufficiently rich, that every element of some identity neighbourhood belongs to a $1$-parameter subgroup.

math.GR↗

arXiv2016

Coarse equivalence and topological couplings of locally compact groups

A result due to M. Gromov states that any two finitely generated groups Γ and Λ are quasi-isometric if and only if they admit a topological coupling, i.e., a commuting pair of proper continuous cocompact actions $Γ\curvearrowright X\curvearrowleft Λ$ on a locally compact Hausdorff space. This result is extended here to all (compactly generated) locally compact second countable groups.

math.GR↗

arXiv2016

Equivariant geometry of Banach spaces and topological groups

We study uniform and coarse embeddings between Banach spaces and topological groups. A particular focus is put on equivariant embeddings, i.e., continuous cocycles associated to continuous affine isometric actions of topological groups on separable Banach spaces with varying geometry.

math.FA↗