Simple locally compact groups acting on trees and their germs of automorphisms

Automorphism groups of locally finite trees provide a large class of examples of simple totally disconnected locally compact groups. It is desirable to understand the connections between the global and local structure of such a group. Topologically, the local structure is given by the commensurability class of a vertex stabiliser; on the other hand, the action on the tree suggests that the local structure should correspond to the local action of a stabiliser of a vertex on its neighbours. We study the interplay between these different aspects for the special class of groups satisfying Tits' independence property. We show that such a group has few open subgroups if and only if it acts locally primitively. Moreover, we show that it always admits many germs of automorphisms which do not extend to automorphisms, from which we deduce a negative answer to a question by George Willis. Finally, under suitable assumptions, we compute the full group of germs of automorphisms; in some specific cases, these turn out to be simple and compactly generated, thereby providing a new infinite family of examples which generalise Neretin's group of spheromorphisms. Our methods describe more generally the abstract commensurator group for a large family of self-replicating profinite branch groups

A combination theorem for combinatorially non-positively curved complexes of hyperbolic groups

We prove a combination theorem for hyperbolic groups, in the case of groups acting on complexes displaying combinatorial features reminiscent of non-positive curvature. Such complexes include for instance weakly systolic complexes and C'(1/6) small cancellation polygonal complexes. Our proof involves constructing a potential Gromov boundary for the resulting groups and analyzing the dynamics of the action on the boundary in order to use Bowditch's characterization of hyperbolicity. A key ingredient is the introduction of a combinatorial property that implies a weak form of non-positive curvature, and which holds for large classes of complexes. As an application, we study the hyperbolicity of groups obtained by small cancellation over a graph of hyperbolic groups.Comment: final preprint version, to appear in Math. Proc. Cambridge Philos. So

Automatic continuity for homomorphisms into free products

A homomorphism from a completely metrizable topological group into a free product of groups whose image is not contained in a factor of the free product is shown to be continuous with respect to the discrete topology on the range. In particular, any completely metrizable group topology on a free product is discrete.Comment: 15 pages, 1 table. Final version. To appear in the Journal of Symbolic Logi

Applications of infinite-dimensional geometry and Lie theory

Habilitation thesisHabilitationsschriftInfinite-dimensional manifolds and Lie groups arise from problems related to differential geometry, fluid dynamics, and the symmetry of evolution equations. Among the most prominent examples of infinite-dimensional manifolds are manifolds of (differentiable) mappings and the diffeomorphism groups Diff(K), where K is a smooth and compact manifold. The group Diff(K) is an infinite-dimensional Lie group which arises naturally in fluid dynamics if K is a three-dimensional torus. The motion of a particle in the fluid corresponds, under periodic boundary conditions, to a curve in Diff(K). As a working definition, an infinite-dimensional Lie group will be a group which at the same time is an infinite-dimensional manifold that turns the group operations into smooth mappings. An infinite-dimensional manifold will be a topological space which is locally (in charts) homeomorphic to an open subset of an infinite-dimensional space. Moreover, we require the change of charts to be smooth. Beyond the realm of Banach spaces, the usual concept of smoothness is no longer available and we replace it with the requirement that all directional derivatives exist and induce continuous mappings, the so called Bastiani calculus. Infinite-dimensional Lie groups and their homogeneous spaces will be the objects of our main interest. In conjunction with Lie theory, we exploit tools from (infinite-dimensional) Riemannian geometry. Recall that a Riemannian metric on a manifold is a choice of inner product for every tangent space which ”depends smoothly” on the basepoint. Generalising Riemannian geometry to infinite-dimensional manifolds, one faces in general the problem that there are no (smooth) partitions of unity. Further, the inner products will in general not be compatible with the topology of the tangent spaces as they are not Hilbert spaces. Thus the finite-dimensional definition of a Riemannian metric (what we will call a ’strong Riemannian metric’) has to be relaxed to admit relevant examples beyond the Hilbert manifold setting. This leads to the notion of a ’weak Riemannian metric’, i.e. a smooth choice of inner products on each tangent space which do not necessarily induce the topology of the tangent space. Constructing weak Riemannian metrics on manifolds of mappings from the L2-inner product, the resulting metrics are studied for example in shape analysis, fluid dynamics and optimal transport. The present thesis explores structures from infinite-dimensional Lie theory and Riemannian geometry, their interplay and applications in three main topics: - Connections between infinite-dimensional Lie groups and higher geometry, - Hopf algebra character groups as Lie groups, and - Applications of the interplay between Lie theory and Riemannian geometry.publishedVersio

Trees, contraction groups, and Moufang sets

We study closed subgroups $G$ of the automorphism group of a locally finite tree $T$ acting doubly transitively on the boundary. We show that if the stabiliser of some end is metabelian, then there is a local field $k$ such that $\mathrm{PSL}_2(k) \leq G \leq \mathrm{PGL}_2(k)$. We also show that the contraction group of some hyperbolic element is closed and torsion-free if and only if $G$ is (virtually) a rank one simple $p$-adic analytic group for some prime $p$. A key point is that if some contraction group is closed, then $G$ is boundary-Moufang, meaning that the boundary $\partial T$ is a Moufang set. We collect basic results on Moufang sets arising at infinity of locally finite trees, and provide a complete classification in case the root groups are torsion-free.Comment: 24 page

Ideal boundaries of pseudo-Anosov flows and uniform convergence groups, with connections and applications to large scale geometry

Given a general pseudo-Anosov flow in a three manifold, the orbit space of the lifted flow to the universal cover is homeomorphic to an open disk. We compactify this orbit space with an ideal circle boundary. If there are no perfect fits between stable and unstable leaves and the flow is not topologically conjugate to a suspension Anosov flow, we then show: The ideal circle of the orbit space has a natural quotient space which is a sphere and is a dynamical systems ideal boundary for a compactification of the universal cover of the manifold. The main result is that the fundamental group acts on the flow ideal boundary as a uniform convergence group. Using a theorem of Bowditch, this yields a proof that the fundamental group of the manifold is Gromov hyperbolic and it shows that the action of the fundamental group on the flow ideal boundary is conjugate to the action on the Gromov ideal boundary. This implies that pseudo-Anosov flows without perfect fits are quasigeodesic flows and we show that the stable/unstable foliations of these flows are quasi-isometric. Finally we apply these results to foliations: if a foliation is R-covered or with one sided branching in an atoroidal three manifold then the results above imply that the leaves of the foliation in the universal cover extend continuously to the sphere at infinity.Comment: 69 pages. Major revision, more explanations and simplified some simplified proofs. Detailed explanations of scalloped regions, parabolic points and perfect fit horoballs. 22 figures (3 new figures