Polish Group Topology Research Articles

Simplicity, bounded normal generation, and automatic continuity of groups of unitaries

We show that the commutator subgroup of the group of unitaries connected to the identity in a simple unital C*-algebra is simple modulo its center. We then go on to investigate the role of “regularity properties” in the structure of the special unitary group of a C*-algebra. Under mild assumptions, we show that this group has the invariant automatic continuity property and a unique polish group topology. Strengthening our assumptions in the case of simple C*-algebras, we show that the special unitary group modulo its center has bounded normal generation. These results apply to all simple purely infinite C*-algebras and to all simple nuclear C*-algebras in the “classifiable class”. We show with counterexamples how our conclusions may in general fail if no regularity conditions are imposed on the C*-algebra.

Polish topologies on groups of non-singular transformations

In this paper, we prove several results concerning Polish group topologies on groups of non-singular transformation. We first prove that the group of measure-preserving transformations of the real line whose support has finite measure carries no Polish group topology. We then characterize the Borel $\sigma$-finite measures $\lambda$ on a standard Borel space for which the group of $\lambda$-preserving transformations has the automatic continuity property. We finally show that the natural Polish topology on the group of all non-singular transformations is actually its only Polish group topology.

Polishability of some groups of interval and circle diffeomorphisms

Let $M=I$ or $M=\mathbb{S}^1$ and let $k\geq 1$. We exhibit a new infinite class of Polish groups by showing that each group $\mathop{\rm Diff}_+^{k+AC}(M)$, consisting of those $C^k$ diffeomorphisms whose $k$-th derivative is absolutely continuous, admits a natural Polish group topology which refines the subspace topology inherited from $\mathop{\rm Diff}_+^k(M)$. By contrast, the group $\mathop{\rm Diff}_+^{1+BV}(M)$, consisting of $C^1$ diffeomorphisms whose derivative has bounded variation, admits no Polish group topology whatsoever.

Polish topologies for graph products of groups

We give strong necessary conditions on the admissibility of a Polish group topology for an arbitrary graph product of groups $G(\Gamma, G_a)$, and use them to give a characterization modulo a finite set of nodes. As a corollary, we give a complete characterization in case all the factor groups $G_a$ are countable.

Bounded normal generation and invariant automatic continuity

We study the question of how quickly products of a fixed conjugacy class in the projective unitary group of a II1-factor von Neumann algebra cover the entire group. Our result is that the number of factors that are needed is essentially as small as permitted by the 1-norm – in analogy to results of Liebeck and Shalev for non-abelian finite simple groups. As an application of the techniques, we prove that every homomorphism from the projective unitary group of a finite factor to a Polish SIN group is continuous – a result which is even new for PU(n). Moreover, we show that the projective unitary group of a II1-factor carries a unique Polish group topology.

Model Theory Methods for Topological Groups

This thesis gathers different works approaching subjects of topological dynamics by means of logic and descriptive set theory, and conversely. The first part is devoted to the study of Roelcke precompact Polish groups, which are the same as the automorphism groups of $\aleph_0$-categorical structures. They form a rich family of examples of infinite-dimensional topological groups, including several interesting permutation groups, isometry groups and homeomorphism groups of distinguished mathematical objects. Building on previous work of Ben Yaacov and Tsankov, we develop a model-theoretic translation of several dynamical aspects of these groups. Then we use this translation to obtain a precise understanding, in this case, of the dynamical hierarchy studied by Glasner and Megrelishvili. Later, with I. Ben Yaacov and T. Tsankov, we provide a model-theoretic description of the Hilbert-compactification of oligomorphic groups, and we give a characterization of Eberlein oligomorphic groups. We also study automorphism groups of randomized structures, as well the separable models of the theory of beautiful pairs of randomizations. The second part, with J. Melleray, studies full groups of minimal homeomorphisms of the Cantor space and their invariant measures. We show that full groups of minimal homeomorphisms do not admit a Polish group topology, and are moreover non-Borel subsets of the homeomorphism group of the Cantor space. We then study the closures of full groups by means of Fraisse theory. Finally, we give a characterization of the sets of invariant measures of minimal homeomorphisms of the Cantor space.

Polish topologies for graph products of cyclic groups

We give a complete characterization of the graph products of cyclic groups admitting a Polish group topology, and show that they are all realizable as the group of automorphisms of a countable structure. In particular, we characterize the right-angled Coxeter groups (resp. Artin groups) admitting a Polish group topology. This generalizes results from [8], [9] and [4].

On a measurable analogue of small topological full groups

We initiate the study of a measurable analogue of small topological full groups that we call L1 full groups. These groups are endowed with a Polish group topology which admits a natural complete right invariant metric. We mostly focus on L1 full groups of measure-preserving Z-actions which are actually a complete invariant of flip conjugacy.We prove that for ergodic actions the closure of the derived group is topologically simple although it can fail to be simple. We also show that the closure of the derived group is connected, and that for measure-preserving free actions of non-amenable groups the closure of the derived group and the L1 full group itself are never amenable.In the case of a measure-preserving ergodic Z-action, the closure of the derived group is shown to be the kernel of the index map. If such an action is moreover by homeomorphism on the Cantor space, we show that the topological full group is dense in the L1 full group. Using Juschenko–Monod and Matui's results on topological full groups, we conclude that L1 full groups of ergodic Z-actions are amenable as topological groups, and that they are topologically finitely generated if and only if the Z-action has finite entropy.

Group metrics for graph products of cyclic groups

We complement the characterization of the graph products of cyclic groups G(Γ,p) admitting a Polish group topology of [9] with the following result. Let G=G(Γ,p), then the following are equivalent:(i)there is a metric on Γ which induces a separable topology in which EΓ is closed;(ii)G(Γ,p) is embeddable into a Polish group;(iii)G(Γ,p) is embeddable into a non-Archimedean Polish group. We also construct left-invariant separable group ultrametrics for G=G(Γ,p) and Γ a closed graph on the Baire space, which is of independent interest.

AUTOMATIC CONTINUITY FOR ISOMETRY GROUPS

We present a general framework for automatic continuity results for groups of isometries of metric spaces. In particular, we prove automatic continuity property for the groups of isometries of the Urysohn space and the Urysohn sphere, i.e. that any homomorphism from either of these groups into a separable group is continuous. This answers a question of Ben Yaacov, Berenstein and Melleray. As a consequence, we get that the group of isometries of the Urysohn space has unique Polish group topology and the group of isometries of the Urysohn sphere has unique separable group topology. Moreover, as an application of our framework we obtain new proofs of the automatic continuity property for the group $\text{Aut}([0,1],\unicode[STIX]{x1D706})$, due to Ben Yaacov, Berenstein and Melleray and for the unitary group of the infinite-dimensional separable Hilbert space, due to Tsankov.

More Polish full groups

We associate to every action of a Polish group on a standard probability space a Polish group that we call the orbit full group. For discrete groups, we recover the well-known full groups of pmp equivalence relations equipped with the uniform topology. However, there are many new examples, such as orbit full groups associated to measure-preserving actions of locally compact groups. We also show that such full groups are complete invariants of orbit equivalence.We give various characterizations of the existence of a dense conjugacy class for orbit full groups, and we show that the ergodic ones actually have a unique Polish group topology. Furthermore, we characterize ergodic full groups of countable pmp equivalence relations as those admitting non-trivial continuous character representations.

Full groups of minimal homeomorphisms and Baire category methods

We study full groups of minimal actions of countable groups by homeomorphisms on a Cantor space$X$, showing that these groups do not admit a compatible Polish group topology and, in the case of$\mathbb{Z}$-actions, are coanalytic non-Borel inside$\text{Homeo}(X)$. We point out that the full group of a minimal homeomorphism is topologically simple. We also study some properties of the closure of the full group of a minimal homeomorphism inside$\text{Homeo}(X)$.

Characterizable groups: Some results and open questions

Let X be an Abelian topological group and X∧ its dual group. A subgroup H of X is called characterized if there is a sequence {un} in X∧ such that H={x∈X:(un,x)→1}. A Polish Abelian group G is called characterizable if there is a continuous monomorphism p from G into a compact metrizable Abelian group X with dense image such that p(G) is a characterized subgroup of X. Every characterizable group is locally quasi-convex. We prove that every second countable locally compact Abelian group X is characterizable. Thus, every second countable locally compact Abelian group is the dual group of a complete countable maximally almost periodic group. It is shown that each characterizable Abelian group of finite exponent is locally compact. Analogously to the Abelian case, we define characterized subgroups of non-Abelian compact metrizable groups and non-Abelian characterizable groups. Using the ℓp-sum of metric groups with two-sided invariant metrics, it is proved that every characterized subgroup admits a Polish group topology.

On self-similarities of ergodic flows

Given an ergodic flow $T=(T_t)_{t\in\Bbb R}$, let $I(T)$ be the set of reals $s\ne 0$ for which the flows $(T_{st})_{t\in\Bbb R}$ and $T$ are isomorphic. It is proved that $I(T)$ is a Borel subset of $\Bbb R^*$. It carries a natural Polish group topology which is stronger than the topology induced from $\Bbb R$. There exists a mixing flow $T$ such that $I(T)$ is an uncountable meager subset of $\Bbb R^*$. For a generic flow $T$, the transformations $T_{t_1}$ and $T_{t_2}$ are spectrally disjoint whenever $|t_1|\ne |t_2|$. A generic transformation (i) embeds into a flow $T$ with $I(T)=\{1\}$ and (ii) does not embed into a flow with $I(T)\ne \{1\}$. For each countable multiplicative subgroup $S\subset\Bbb R^*$, it is constructed a Poisson suspension flow $T$ with simple spectrum such that $I(T)=S$. If $S$ is without rational relations then there is a rank-one weakly mixing rigid flow $T$ with $I(T)=S$.

Topology of the isometry group of the Urysohn space

Using classical results of infinite-dimensional geometry, we show that the isometry group of the Urysohn space, endowed with its usual Polish group topology, is homeomorphic to the separable Hilbert space $\ell^2({\mathbb N})$. The proof is based on a lem

The coset equivalence relation and topologies on subgroups

The paper studies the structure of the homogeneous space $G/H$, for $G$ a Polish group and $H< G$ a Borel, not necessarily closed subgroup of $G$, from the point of view of the theory of definable equivalence relations. It makes a connection between the complexity of the natural {\it coset equivalence relation} associated with $G/H$ and {\it Polishability} of $H$, that is, the possibility of introducing a Polish group topology on $H$ respecting its Borel structure. In particular, it is proved that if $H$ is an Abelian Borel subgroup of a Polish group $G$, then either $H$ is Polishable or ${\Bbb E}_1$ continuously embeds into the coset equivalence relation induced by $H$ on $G$. The same conclusion is shown to hold if $H$ is an increasing union of a sequence of Polishable subgroups of $G$.

Uniqueness of Polish group topology

We show the limits of Mackey's theorem applied to identity sets to prove that a given group has a unique Polish group topology. Verbal sets in Abelian Polish groups and full verbal sets in the infinite symmetric group are Borel. However this is not true in general. A Polish group with a neighborhood π-base at 1 of sets from the σ-algebra of identity and verbal sets has a unique Polish group topology. It follows that compact, connected, simple Lie groups, as well as finitely generated profinite groups, have a unique Polish group topology.

On the non-existence of certain group topologies

Minimal Hausdorff (Baire) group topologies of certain groups of transformations naturally occurring in analysis are studied. The results obtained are subsequently applied to show that, e.g., the homeomorphism groups of the rational and of the irrational numbers carry no Polish group topology. In answer to a question of A.S. Kechris it is shown that that the group of Borel automorphisms of R cannot be a Polish group either.