Related papers: Limits of contraction groups and the Tits core

Associated to any closed subgroup $G\subset U_N^+$ is a family of toral subgroups $T_Q\subset G$, indexed by the unitary matrices $Q\in U_N$. The family $\{T_Q|Q\in U_N\}$ is expected to encode the main properties of $G$, and there are…

Let k be a local field, and G a linear group over k. We prove that either G contains a relatively open solvable subgroup, or it contains a relatively dense free subgroup. This result has applications in dynamics, Riemannian foliations and…

Call a compact, connected, simple Lie group $G$ {\emph{adjoint simple}} if it has trivial center. Let $C\subset G$ be a nontrivial conjugacy class, $e\in G$ the identity element of $G$. We prove the existence of an $N\in\mathbb{N}$,…

A subgroup $H$ of a finite group $G$ is said to be an $\mathscr{H}C$-subgroup of $G$ if there exists a normal subgroup $T$ of $G$ such that $G=HT$ and $H^g \cap N_T(H)\leq H$ for all $g\in G$. In this paper, we investigate the structure of…

We determine the groups which can appear as the normalizer of a maximal torus in a connected 2-compact group. The technique depends on using ideas of Tits to give a novel description of the normalizer of the torus in a connected compact Lie…

It is shown that a group defined by forbidding all patterns of size s+1 that do not appear in a given self-similar group of tree automorphisms is the topological closure of a self-similar, countable, regular branch group, branching over its…

Let $G$ be a connected closed subgroup of $\mathrm{GL}_n(\mathbb{C})$ which is simple as a Lie group and which acts irreducibly on $\mathbb{C}^n$. Regarding both $G$ and its Lie algebra $\mathfrak{g}$ as subsets of $M_n(\mathbb{C})$, we…

We observe a correspondence between collections of closed subgroups and normal subgroups in totally disconnected locally compact groups. This correspondence is applied to prove structure theorems for two classes of totally disconnected…

Let G be a group acting geometrically on a CAT(0) cube complex X. We prove first that G is hyperbolic relative to the collection P of subgroups if and only if the simplicial boundary of X is the disjoint union of a nonempty discrete set,…

For any topological group $G$ the dual object $\hat G$ is defined as the set of equivalence classes of irreducible unitary representations of $G$ equipped with the Fell topology. If $G$ is compact, $\hat G$ is discrete. In an earlier paper…

We define Cartan subgroups in connected locally compact groups, which extends the classical notion of Cartan subgroups in Lie groups. We prove their existence and justify our choice of the definition which differs from the one given by…

We study contraction groups for automorphisms of totally disconnected locally compcat groups using the scale of the automorphism as a tool. The contraction group is shown to be unbounded when the inverse automorphism has non-trivial scale…

We show that recent results of U. Baumgartner and G.A. Willis concerning contraction groups of automorphisms of metrizable, totally disconnected, locally compact groups remain valid also in the non-metrizable case, if one restricts…

We study topological groups having all closed subgroups (totally) minimal and we call such groups c-(totally) minimal. We show that a locally compact c-minimal connected group is compact. Using a well-known theorem of Hall and Kulatilaka…

If a discrete subset S of a topological group G with the identity 1 generates a dense subgroup of G and S \cup {1} is closed in G, then S is called a suitable set for G. We apply Michael's selection theorem to offer a direct,…

A group is said to be C*-simple if its reduced C*-algebra is simple. We establish an intrinsic (group-theoretic) characterization of groups with this property. Specifically, we prove that a discrete group is C*-simple if and only if it has…

There are strong analogies between groups definable in o-minimal structures and real Lie groups. Nevertheless, unlike the real case, not every definable group has maximal definably compact subgroups. We study definable groups G which are…

A simple Steinberg algebra associated to an ample Hausdorff groupoid $G$ is algebraically purely infinite if and only if the characteristic functions of compact open subsets of the unit space are infinite idempotents. If a simple Steinberg…

We prove the Tits alternative for an almost coherent $PD(3)$ group which is not virtually properly locally cyclic. In particular, we show that an almost coherent $PD(3)$ group which cannot be generated by fewer than four elements always…

We prove an asymptotically tight bound on the extremal density guaranteeing subdivisions of bounded-degree bipartite graphs with a mild separability condition. As corollaries, we answer several questions of Reed and Wood on embedding sparse…