Related papers: Aspects of automatic continuity

We show that the topological full group of a Hausdorff ample groupoid with compact unit space coincides with the group of homotopy classes of invertible isometries in pseudofunction algebras associated with the groupoid. Moreover, if the…

We study the automorphism group of graphons (graph limits). We prove that after an appropriate "standardization" of the graphon, the automorphism group is compact. Furthermore, we characterize the orbits of the automorphism group on…

We develop an explicit covering theory for complexes of groups, parallel to that developed for graphs of groups by Bass. Given a covering of developable complexes of groups, we construct the induced monomorphism of fundamental groups and…

We prove twisted homological stability with polynomial coefficients for automorphism groups of free nilpotent groups of any given class. These groups interpolate between two extremes for which homological stability was known before, the…

This survey describes some useful properties of the local homology of abstract simplicial complexes. Although the existing literature on local homology is somewhat dispersed, it is largely dedicated to the study of manifolds, submanifolds,…

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…

In this paper we focus on the set-open topologies on the group $\mathcal{H}(X)$ of all self-homeomorphisms of a topological space $X$ which yield continuity of both the group operations, product and inverse function. As a consequence, we…

We study notions of persistent homotopy groups of compact metric spaces together with their stability properties in the Gromov-Hausdorff sense. We pay particular attention to the case of fundamental groups, for which we obtain a more…

We introduce and study Polish topologies on various spaces of countable enumerated groups, where an enumerated group is simply a group whose underlying set is the set of natural numbers. Using elementary tools and well known examples from…

Necessary and sufficient conditions for the exactness (in the algebraic sense) of certain sequences of continuous group homomorphisms are established.

We prove that the homeomorphism problem for connected compact metric spaces is Borel bireducible with a universal orbit equivalence relation induced by a Borel action of a Polish group.

A topological space is almost locally compact if it contains a dense locally compact subspace. We generalize a result from \cite{Ma}, showing that isomorphism on Borel classes of almost locally compact Polish metric structures is always…

We construct some new cohomology theories for topological groups and Lie groups and study some of its basic properties. For example, we introduce a cohomology theory based on measurable cochains which are continuous in a neighbourhood of…

We show that every abstract homomorphism $\varphi$ from a locally compact group $L$ to a graph product $G_\Gamma$, endowed with the discrete topology, is either continuous or $\varphi(L)$ lies in a 'small' parabolic subgroup. In particular,…

Assume that there is a free group action of automorphisms on a bipartite graph. If there is a perfect matching on the factor graph, then obviously there is a perfect matching on the graph. Surprisingly, the reversed is also true for…

This paper aims to examine the version of the topological group structure in proximity and especially descriptive proximity spaces, that is, the concepts of proximal group and descriptive proximal group are introduced. In addition, the…

In this note, we outline the general development of a theory of symmetric homology of algebras, an analog of cyclic homology where the cyclic groups are replaced by symmetric groups. This theory is developed using the framework of crossed…

We study locally compact group topologies on semisimple Lie groups. We show that the Lie group topology on such a group $S$ is very rigid: every 'abstract' isomorphism between $S$ and a locally compact and $\sigma$-compact group $\Gamma$ is…

In an earlier paper, we introduced the following pre-order on the subgroups of a given Polish group: if $G$ is a Polish group and $H,L \subseteq G$ are subgroups, we say $H$ is {\em homomorphism reducible} to $L$ iff there is a continuous…

For any smooth compact manifold $W$ of dimension at least two we prove that the classifying spaces of its group of diffeomorphisms which fix a set of $k$ points or $k$ embedded disks (up to permutation) satisfy homology stability. The same…