### A combinatorial proof of the extension property for partial isometries

We present a short and self-contained proof of the extension property for partial isometries of the class of all finite metric spaces.

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We present a short and self-contained proof of the extension property for partial isometries of the class of all finite metric spaces.

The Polish space Y constructed in [vM1] admits no nontrivial isotopy. Yet, there exists a Polish group that acts transitively on Y.

Let G be a group which acts by homeomorphisms on a metric space X. We say the action of G is locally moving on X if for every open U ⊆ X there is a g ∈ G such that g↾X ≠ Id while g↾(X∖U) = Id. We prove the following theorem: Theorem A. Let X,Y be completely metrizable spaces and let G be a group which acts on X and Y with locally moving actions. If the orbits of the action of G on X are of the second category in X and the orbits of the action of G on Y are of the second category...

We study a problem of isometric compact 2-step nilmanifolds $M/\Gamma $ using some information on their geodesic flows, where $M$ is a simply connected 2-step nilpotent Lie group with a left invariant metric and $\Gamma $ is a cocompact discrete subgroup of isometries of $M$. Among various works concerning this problem, we consider the algebraic aspect of it. In fact, isometry groups of simply connected Riemannian manifolds can be characterized in a purely algebraic way, namely by normalizers. So, suitable factorization...

We show that the group of type-preserving automorphisms of any irreducible semiregular thick right-angled building is abstractly simple. When the building is locally finite, this gives a large family of compactly generated abstractly simple locally compact groups. Specialising to appropriate cases, we obtain examples of such simple groups that are locally indecomposable, but have locally normal subgroups decomposing non-trivially as direct products, all of whose factors are locally normal.

The main results of the paper are: Proposition 0.1. A group G acting coarsely on a coarse space (X,𝓒) induces a coarse equivalence g ↦ g·x₀ from G to X for any x₀ ∈ X. Theorem 0.2. Two coarse structures 𝓒₁ and 𝓒₂ on the same set X are equivalent if the following conditions are satisfied: (1) Bounded sets in 𝓒₁ are identical with bounded sets in 𝓒₂. (2) There is a coarse action ϕ₁ of a group G₁ on (X,𝓒₁) and a coarse action ϕ₂ of a...

Let $X$ be a building of arbitrary type. A compactification ${\mathcal{C}}_{\mathrm{sph}}\left(X\right)$ of the set ${\text{Res}}_{\mathrm{sph}}\left(X\right)$ of spherical residues of $X$ is introduced. We prove that it coincides with the horofunction compactification of ${\text{Res}}_{\mathrm{sph}}\left(X\right)$ endowed with a natural combinatorial distance which we call the root-distance. Points of ${\mathcal{C}}_{\mathrm{sph}}\left(X\right)$ admit amenable stabilisers in $\text{Aut}\left(X\right)$ and conversely, any amenable subgroup virtually fixes a point in ${\mathcal{C}}_{\mathrm{sph}}\left(X\right)$. In addition, it is shown that, provided $\text{Aut}\left(X\right)$ is transitive enough, this compactification also coincides with the group-theoretic...

We prove some extension theorems involving uniformly continuous maps of the universal Urysohn space. As an application, we prove reconstruction theorems for certain groups of autohomeomorphisms of this space and of its open subsets.

We show that, whenever $\Gamma $ is a countable abelian group and $\Delta $ is a finitely-generated subgroup of $\Gamma $, a generic measure-preserving action of $\Delta $ on a standard atomless probability space $(X,\mu )$ extends to a free measure-preserving action of $\Gamma $ on $(X,\mu )$. This extends a result of Ageev, corresponding to the case when $\Delta $ is infinite cyclic.

Let $G$ be a connected real semi-simple Lie group and $H$ a closed connected subgroup. Let $P$ be a minimal parabolic subgroup of $G$. It is shown that $H$ has an open orbit on the flag manifold $G/P$ if and only if it has finitely many orbits on $G/P$. This confirms a conjecture by T. Matsuki.

We study problems concerning the Samuel compactification of the automorphism group of a countable first-order structure. A key motivating question is a problem of Furstenberg and a counter-conjecture by Pestov regarding the difference between $S\left(G\right)$, the Samuel compactification, and $E\left(M\right(G\left)\right)$, the enveloping semigroup of the universal minimal flow. We resolve Furstenberg’s problem for several automorphism groups and give a detailed study in the case of $G={S}_{\infty}$, leading us to define and investigate several new types...