Related papers: Connected Polish groups with ample generics
The present article constitutes the third part of our study of the large scale geometry of metrisable groups, the first two part appearing in the companion paper "Large scale geometry of metrisable groups". In this third part, we present a…
Ando, Matsuzawa, Thom, and T\"ornquist have resolved a problem by Sorin Popa by constructing an example of a Polish group of unitary operators with the strong operator topology, whose left and right uniform structures coincide, but which…
A series of recent papers by Bergfalk, Lupini and Panagiotopoulus developed the foundations of a field known as `definable algebraic topology,' in which classical cohomological invariants are enriched by viewing them as groups with a Polish…
The class of multi-EGS groups is a generalisation of the well-known Grigorchuk-Gupta-Sidki (GGS-)groups. Here we classify branch multi-EGS groups with the congruence subgroup property and determine the profinite completion of all branch…
We study the complexity of the isomorphism relation for various classes of closed subgroups of the group of permutations of the natural numbers. We use the setting of Borel reducibility between equivalence relations on Polish spaces. For…
Given a family of subspaces we investigate existence, quantity and quality of common complements in Hilbert spaces and Banach spaces. In particular we are interested in complements for countable families of closed subspaces of finite…
In first order logic, it is known that you can define a topology so that the countable models of some theory $T$ form a Polish Space (i.e. completely metrizable second countable space). In this paper we use the Baldwin- Boney Relational…
The symmetries of surfaces which can be embedded into the symmetries of the 3-dimensional Euclidean space $\mathbb{R}^3$ are easier to feel by human's intuition. We give the maximum order of finite group actions on $(\mathbb{R}^3, \Sigma)$…
We study the usual notion of Scott rank but in the setting of Polish metric spaces. The signature consists of distance relations: for each rational $q > 0$, there is a relation $R_{<q}(x,y)$ stating that the distance of $x$ and $y $ is less…
Countably infinite groups (with a fixed underlying set) constitute a Polish space $G$ with a suitable metric, hence the Baire category theorem holds in $G$. We study isomorphism invariant subsets of $G$, which we call group properties. We…
In this paper, we show that the arc-connectedness equivalence relation on a Polish subspace of the real plane is an essentially hyperfinite Borel equivalence relation. This result provides the optimal upper bound for such a Borel…
We study a property about Polish inverse semigroups similar to the classical theorem of Pettis about Polish groups. In contrast to what happens with Polish groups, not every Polish inverse semigroup have the Pettis property. We present…
We extend the Becker--Kechris topological realization and change-of-topology theorems for Polish group actions in several directions. For Polish group actions, we prove a single result that implies the original Becker--Kechris theorems, as…
In this note we show that groups with definable generics in a separably closed valued of finite imperfection degree can be embedded into groups definable in their algebraic closure.
We show that finitely presented groups which admit $k$-planar Cayley graphs contain finite-index subgroups with planar Cayley graphs. More generally, we answer a question of Georgakopoulos and Papasoglu in the special case of coarsely…
When $G$ is a Polish group, metrizability of the universal minimal flow has been shown to be a robust dividing line in the complexity of the topological dynamics of $G$. We introduce a class of groups, the CAP groups, which provides a neat…
We solve a long-standing open problem, formulated by Krasner in the 1950's, in the context of Polish (i.e. separable complete) ultrametric spaces by providing a characterization of their isometry groups using suitable forms of generalized…
We complement the characterization of the graph products of cyclic groups $G(\Gamma, \mathfrak{p})$ admitting a Polish group topology of [9] with the following result. Let $G = G(\Gamma, \mathfrak{p})$, then the following are equivalent:…
Tkachenko and Yaschenko [34] characterized the abelian groups G such that all proper unconditionally closed subsets of G are finite, these are precisely the abelian groups G having cofinite Zariski topology (they proved that such a G is…
In this paper we consider non-archimedean abelian Polish groups whose orbit equivalence relations are all Borel. Such groups are called tame. We show that a non-archimedean abelian Polish group is tame if and only if it does not involve…