arXiv subjects
Publications and source records attributed to Christian Rosendal.
We investigate extensions of S. Solecki's theorem on closing off finite partial isometries of metric spaces \cite{solecki1} and obtain the following exact equivalence: any action of a discrete group $Γ$ by isometries of a metric space is finitely approximable if and only if any product of finitely generated subgroups of $Γ$ is closed in the profinite topology on $Γ$.
Given a finitely generated group $Γ$, we study the space ${\rm Isom}(Γ,{\mathbb Q\mathbb U})$ of all actions of $Γ$ by isometries of the rational Urysohn metric space ${\mathbb Q\mathbb U}$, where ${\rm Isom}(Γ,{\mathbb Q\mathbb U})$ is equipped with the topology it inherits seen as a closed subset of ${\rm Isom}({\mathbb Q\mathbb U})^Γ$. When $Γ$ is the free group $\F_n$ on $n$ generators this space is just ${\rm Isom}({\mathbb Q\mathbb U})^n$, but is in general significantly more complicated. We prove that when $Γ$ is finitely generated Abelian there is a generic point in ${\rm Isom}(Γ,{\mathbb Q\mathbb U})$, i.e., there is a comeagre set of mutually conjugate isometric actions of $Γ$ on ${\mathbb Q\mathbb U}$.
It is proved that any countable index, universally measurable subgroup of a Polish group is open. By consequence, any universally measurable homomorphism from a Polish group into the infinite symmetric group $S_\infty$ is continuous. It is also shown that a universally measurable homomorphism from a Polish group into a second countable, locally compact group is necessarily continuous.
We consider actions of completely metrisable groups on simplicial trees in the context of the Bass--Serre theory. Our main result characterises continuity of the amplitude function corresponding to a given action. Under fairly mild conditions on a completely metrisable group $G$, namely, that the set of elements generating a non-discrete or finite subgroup is somewhere dense, we show that in any decomposition as a free product with amalgamation, $G=A\ast_CB$, the amalgamated groups $A$, $B$ and $C$ are open in $G$.
We construct a family of separable Hilbertian operator spaces, such that the relation of complete isomorphism between the subspaces of each member of this family is complete $\ks$. We also investigate some interesting properties of completely unconditional bases of the spaces from this family. In the Banach space setting, we construct a space for which the relation of isometry of subspaces is equivalent to equality of real numbers.
We study infinite asymptotic games in Banach spaces with an F.D.D. and prove that analytic games are determined by characterising precisely the conditions for the players to have winning strategies. These results are applied to characterise spaces embeddable into $\ell_p$ sums of finite dimensional spaces, extending results of Odell and Schlumprecht, and to study various notions of homogeneity of bases and Banach spaces. These results are related to questions of rapidity of subsequence extraction from normalised weakly null sequences.
We prove an exact, i.e., formulated without $Δ$-expansions, Ramsey principle for infinite block sequences in vector spaces over countable fields, where the two sides of the dichotomic principle are represented by respectively winning strategies in Gowers' block sequence game and winning strategies in the infinite asymptotic game. This allows us to recover Gowers' dichotomy theorem for block sequences in normed vector spaces by a simple application of the basic determinacy theorem for infinite asymptotic games.
It is proved that the relation of isomorphism between separable Banach spaces is a complete analytic equivalence relation, i.e., that any analytic equivalence relation Borel reduces to it. Thus, separable Banach spaces up to isomorphism provide complete invariants for a great number of mathematical structures up to their corresponding notion of isomorphism. The same is shown to hold for (1) complete separable metric spaces up to uniform homeomorphism, (2) separable Banach spaces up to Lipschitz isomorphism, and (3) up to (complemented) biembeddability, (4) Polish groups up to topological isomorphism, and (5) Schauder bases up to permutative equivalence. Some of the constructions rely on methods recently developed by S. Argyros and P. Dodos.
We study topological properties of conjugacy classes in Polish groups, with emphasis on automorphism groups of homogeneous countable structures. We first consider the existence of dense conjugacy classes (the topological Rokhlin property). We then characterize when an automorphism group admits a comeager conjugacy class (answering a question of Truss) and apply this to show that the homeomorphism group of the Cantor space has a comeager conjugacy class (answering a question of Akin-Hurley-Kennedy). Finally, we study Polish groups that admit comeager conjugacy classes in any dimension (in which case the groups are said to admit ample generics). We show that Polish groups with ample generics have the small index property (generalizing results of Hodges-Hodkinson-Lascar-Shelah) and arbitrary homomorphisms from such groups into separable groups are automatically continuous. Moreover, in the case of oligomorphic permutation groups, they have uncountable cofinality and the Bergman property. These results in particular apply to automorphism groups of many $ω$-stable, $\aleph_0$-categorical structures and of the random graph. In this connection, we also show that the infinite symmetric group $S_\infty$ has a unique non-trivial separable group topology. For several interesting groups we also establish Serre's properties (FH) and (FA).
We prove that arbitrary homomorphisms from one of the groups ${\rm Homeo}(\ca)$, ${\rm Homeo}(\ca)^\N$, ${\rm Aut}(\Q,<)$, ${\rm Homeo}(\R)$, or ${\rm Homeo}(S^1)$ into a separable group are automatically continuous. This has consequences for the representations of these groups as discrete groups. For example, it follows, in combination with a result on V.G. Pestov, that any action of the discrete group ${\rm Homeo}_+(\R)$ by homeomorphisms on a compact metric space has a fixed point.
We consider the unitary group $\U$ of complex, separable, infinite-dimensional Hilbert space as a discrete group. It is proved that, whenever $\U$ acts by isometries on a metric space, every orbit is bounded. Equivalently, $\U$ is not the union of a countable chain of proper subgroups, and whenever $\E\subseteq \U$ generates $\U$, it does so by words of a fixed finite length.
We show that any homomorphism from the homeomorphism group of a compact 2-manifold, with the compact-open topology, or equivalently, with the topology of uniform convergence, into a separable topological group is automatically continuous.
A topological group G is defined to have property (OB) if any G-action by isometries on a metric space, which is separately continuous, has bounded orbits. We study this topological analogue of the socalled Bergman property in the context of Polish groups, where we show it to have several interesting reformulations and consequences. We subsequently apply the results obtained in order to verify property (OB) for a number of groups of isometries and homeomorphism groups of compact metric spaces. We also give a proof that the isometry group of the rational Urysohn metric space of diameter 1 is Bergman.
A Banach space contains either a minimal subspace or a continuum of incomparable subspaces. General structure results for analytic equivalence relations are applied in the context of Banach spaces to show that if $E_0$ does not reduce to isomorphism of the subspaces of a space, in particular, if the subspaces of the space admit a classification up to isomorphism by real numbers, then any subspace with an unconditional basis is isomorphic to its square and hyperplanes and has an isomorphically homogeneous subsequence.
We show that any Banach space contains a continuum of non isomorphic subspaces or a minimal subspace. We define an ergodic Banach space $X$ as a space such that $E_0$ Borel reduces to isomorphism on the set of subspaces of $X$, and show that every Banach space is either ergodic or contains a subspace with an unconditional basis $ which is complementably universal for the family of its block-subspaces. We also use our methods to get uniformity results; for example, in combination with a result of B. Maurey, V. Milman and N. Tomczak-Jaegermann, we show that an unconditional basis of a Banach space, of which every block-subspace is complemented, must be asymptotically $c_0$ or $l_p$.
We consider a normalized basis in a Banach space with the following property: any normalized block sequence of the basis has a subsequence equivalent to the basis. We show that under uniformity or other natural assumptions, a basis with this property is equivalent to the unit vector basis of $c_0$ or $\ell_p$. We also address an analogous problem for spreading models.
If a Banach space has an unconditional basis it either contains a continuum of non isomorphic subspaces or is isomorphic to its square and hyperplanes and satisfies other regularity properties. An HI Banach space contains a continuum of non isomorphic subspaces.