In the present paper we characterize the surjective isometries of the space of compact, convex subsets of proper, geodesically complete CAT(0)-spaces in which geodesics do not split, endowed with the Hausdorff metric. Moreover, an analogue characterization of the surjective isometries of the space of compact subsets of a proper, uniquely geodesic, geodesically complete metric space in which geodesics do not split, when endowed with the Hausdorff metric, is given.
On any convex domain in $\mathbb{R}^n$ we can define the Hilbert metric. A projective transformation is an example of an isometry of the Hilbert metric. In this thesis we will prove that the group of projective transformations on a convex domain has at most index 2 in the group of isometries of the convex domain with its Hilbert metric. Furthermore we will give criteria for which the set of projective transformations between two convex domains is equal to the set of isometries of the Hilbert metric of these convex domains. Lastly we will show that $2$-dimensional convex domains with their corr…
In this paper, we show that the $C^1$-differentiability of the norm of a two-dimensional normed space depends only on distances between points of the unit sphere in two different ways. As a consequence, we see that any isometry between the spheres of normed planes $τ:S_X\to S_Y$ is linear, provided that there exist linearly independent $x,\overline{x}\in S_X$ where $S_X$ is not differentiable and that $S_X$ is piecewise differentiable. We end this work by showing that the isometry $τ:C_X\to C_Y$ is linear even if it is not an isometry between spheres: every isometry between (planar) Jordan pie…
We introduce a class of functional analogs of the symmetric difference metric on the space of coercive convex functions on $\mathbb{R}^n$ with full-dimensional domain. We show that convergence with respect to these metrics is equivalent to epi-convergence. Furthermore, we give a full classification of all isometries with respect to some of the new metrics. Moreover, we introduce two new functional analogs of the Hausdorff metric on the spaces of coercive convex functions and super-coercive convex functions, respectively, and prove equivalence to epi-convergence.
In this paper we study convex subcomplexes of spherical buildings. We pay special attention to fixed point sets of type-preserving isometries of spherical buildings. This sets are also convex subcomplexes of the natural polyhedral structure of the building. We show, among other things, that if the fixed point set is top-dimensional then it is either a subbuilding or it has circumradius $\leq \fracπ{2}$. If the building is of type $A_n$ or $D_n$, we also show that the same conclusion holds for an arbitrary (top-dimensional in the $D_n$-case) convex subcomplex. This proves a conjecture of Kleine…
Let G be a non-elementary group of isometries of a proper CAT(0)-space with limit set L. We survey properties of the action of G on L under the assumption that G contains a rank-one element. Among others, we show that there is a dense orbit for the action of G on the complement of the diagonal in LxL and that pairs of fixed points of rank-one elements are dense in the complement of the diagonal of LxL.
We study the fixed point set in the ideal boundary of a parabolic isometry of a proper CAT(0)-space. We show that the radius of the fixed point set is at most pi/2, and study its centers. As a consequence, we prove that the set of fixed points is contractible with respect to the Tits topology.
One of the basic problems in discrete geometry is to determine the most efficient packing of congruent replicas of a given convex set $K$ in the plane or in space. The most commonly used measure of efficiency is density. Several types of the problem arise depending on the type of isometries allowed for the packing: packing by translates, lattice packing, translates and point reflections, or all isometries. Due to its connections with number theory, crystallography, etc., lattice packing has been studied most extensively. In two dimensions the theory is fairly well developed, and there are seve…
We give positive answers for questions by Berestovskii. Namely, we prove that every bijection of locally compact geodesically complete and connected at infinity CAT(0)-space $X$ onto itself preserving some fixed distance or satellite relations is an isometry of this space. The proof of this theorem is based on another result stated by Berestovskii as a problem: the metric of the space $X$ may be recovered from its diagonal tube corresponding to an arbitrary number $r > 0$.
In this paper we extend results by De la Harpe concerning the isometries of strictly convex Hilbert geometries, and the characterisation of the isometry groups of Hilbert geometries on finite dimensional simplices, to infinite dimensions. The proofs rely on a mix of geometric and functional analytic methods.
We consider a `contracting boundary' of a proper geodesic metric space consisting of equivalence classes of geodesic rays that behave like geodesics in a hyperbolic space. We topologize this set via the Gromov product, in analogy to the topology of the boundary of a hyperbolic space. We show that when the space is not hyperbolic, quasi-isometries do not necessarily give homeomorphisms of this boundary. Continuity can fail even when the spaces are required to be CAT(0). We show this by constructing an explicit example.
A Fuchsian polyhedron in hyperbolic space is a polyhedral surface invariant under the action of a Fuchsian group of isometries (i.e. a group of isometries leaving globally invariant a totally geodesic surface, on which it acts cocompactly). The induced metric on a convex Fuchsian polyhedron is isometric to a hyperbolic metric with conical singularities of positive singular curvature on a compact surface of genus greater than one. We prove that these metrics are actually realised by exactly one convex Fuchsian polyhedron (up to global isometries). This extends a famous theorem of A.D. Alexandro…
In this paper, we study bijections on strictly convex sets of $\mathbf R \mathbf P^n$ for $n \geq 2$ and closed convex projective surfaces equipped with the Hilbert metric that map complete geodesics to complete geodesics as sets. Hyperbolic $n$-space with its standard metric is a special example of the spaces we consider, and it is known that these bijections in this context are precisely the isometries. We first prove that this result generalizes to an arbitrary strictly convex set. For the surfaces setting, we prove the equivalence of mapping simple closed geodesics to simple closed geodesi…
Let $X$ be a proper, non-compact CAT(-1) space, and $Γ$ a discrete cocompact subgroup of the isometries of $X$. We compactify the diagonal action of $Γ$ on $X \times X$ considering a domain of the horofunction boundary with respect to the maximum metric.
We provide an analysis of the dynamics of isometries and semicontractions of metric spaces. Certain subsets of the boundary at infinity play a fundamental role and are identified completely for the standard boundaries of CAT(0)-spaces, Gromov hyperbolic spaces, Hilbert geometries, certain pseudoconvex domains, and partially for Thurston's boundary of Teichmueller spaces. We present several rather general results concerning groups of isometries, as well as the proof of other more specific new theorems, for example concerning the existence of free nonabelian subgroups in CAT(0)-geometry, iterati…
We give a generalization to convex co-compact semigroups of a beautiful theorem of Patterson-Sullivan, telling that the critical exponent (that is the exponential growth rate) equals the Hausdorff dimension of the limit set (that is the smallest closed non-empty invariant subset), for a isometries discrete group of a proper hyperbolic space with compact boundary. To do that, we introduce a notion of entropy, which generalize the notion of critical exponent of discrete groups, and we show that it is equal to the upper bound of critical exponents of Schottky sub-semigroups (which are semigroups …
We study groups of isometries of packed, geodesically complete, CAT$(0)$-spaces for which the systole at every point is smaller than a universal constant depending only on the packing, deducing strong rigidity results. We show that if a space as above has some negative curvature behaviour then it cannot support a thin action: this generalizes the classical Margulis Lemma to a broader class of spaces.
Let $X$ be a CAT(0) space, and $G$ a discrete cyclic group of isometries of $X$. We investigate the domain of discontinuity for the action of $G$ on the boundary $\partial X$.
Let G be a compact group acting on a Banach space L by means of linear isometries. The action of G on L induces a natural continuous action on cc(L), the hyperspace of all compact convex subsets of L endowed with the Hausdorff metric topology. The main result of this paper states that the G-space cc(L) is a G-AE. Under some extra assumptions, this result can be extended to CB(L), the hyperspace of all closed and bounded convex subsets of L.
We investigate CAT(0) metric spaces whose associated Tits boundary is compact. Prominent examples of such spaces are of course the euclidean ones. However there exist non trivial geodesically complete CAT(0) spaces with compact Tits boundary that are far from being flat. We show that if the isometry group of such a space acts cocompactly, then the space is "as flat as possible", i.e. the boundary minimal subspace must be flat. Our result relies on the existence of hyperbolic isometries, which is a problem itself. We hence add the proof of the following existence proposition: a cocompact CAT(0)…
Heinrich Tietze has shown that for a closed connected subset of euclidean space being convex is a local property. We generalize this to CAT(0)-spaces and locally compact CAT(κ) spaces. As an application we give a construction of certain convex sets in euclidean buildings.
In this note we show that a connected, closed and locally convex subset (with an extra assumption on the diameter with respect to the induced length metric if $κ>0$) of a $CAT(κ)$ space is convex.
We study the closure of the convex hull of a compact set in a complete CAT(0) space. First we give characterization results in terms of compact sets and the closure of their convex hulls for locally compact CAT(0) spaces that are either regular or satisfy the geodesic extension property. Later inspired by a geometric interpretation of Carathéodory's Theorem we introduce the operation of threading for a given set. We show that threading exhibits certain monotonicity properties with respect to intersection and union of sets. Moreover threading preserves compactness. Next from the commutativity o…
We consider actions of locally compact groups $G$ on certain CAT(0) spaces $X$ by isometries. The CAT(0) spaces we consider have finite dimension at large scale. In case $B$ is a $G$-boundary, that is a measurable $G$-space with amenability and ergodicity properties, we prove the existence of equivariant maps from $B$ to the visual boundary $\partial X$.
We give a criterion for a set of $n$ hyperbolic isometries of a $\mathrm{CAT}(0)$ metric space $X$ to generate a free group on $n$ generators. This extends a result by Alperin, Farb and Noskov who proved this for 2 generators under the additional assumption that $X$ is complete and has no fake zero angles. Moreover, when $X$ is locally compact, the group we obtain is also discrete.
For n greater or equal 4 we discuss questions concerning global fixed points for isometric actions of Aut(Fn), the automorphism group of a free group of rank n, on complete CAT(0) spaces. We prove that whenever Aut(Fn) acts by isometries on complete d-dimensional CAT(0) space with d is less than 2 times the integer function of n over 4 and minus 1, then it must fix a point. This property has implications for irreducible representations of Aut(Fn), which are also presented here. For SAut(Fn), the unique subgroup of index two in Aut(Fn), we obtain similar results.
A general fixed point theorem for isometries in terms of metric functionals is proved under the assumption of the existence of a conical bicombing. It is new even for isometries of Banach spaces as well as for non-locally compact CAT(0)-spaces and injective spaces. Examples of actions on non-proper CAT(0)-spaces come from the study of diffeomorphism groups, birational transformations, and compact Kähler manifolds. A special case of the fixed point theorem provides a novel mean ergodic theorem that in the Hilbert space case implies von Neumann's theorem. The theorem accommodates classically fix…
We show that any filtering family of closed convex subsets of a finite-dimensional CAT(0) space $X$ has a non-empty intersection in the visual bordification $ \bar{X} = X \cup \partial X$. Using this fact, several results known for proper CAT(0) spaces may be extended to finite-dimensional spaces, including the existence of canonical fixed points at infinity for parabolic isometries, algebraic and geometric restrictions on amenable group actions, and geometric superrigidity for non-elementary actions of irreducible uniform lattices in products of locally compact groups.
Let $X$ be a proper CAT($0$) space and $G$ a cocompact group of isometries of $X$ without fixed point at infinity. We prove that if $\partial X$ contains an invariant subset of circumradius $π/2$, then $X$ contains a quasi-dense, closed convex subspace that splits as a product. Adding the assumption that the $G$-action on $X$ is properly discontinuous, we give more conditions that are equivalent to a product splitting. In particular, this occurs if $\partial X$ contains a proper nonempty, closed, invariant, $π$-convex set in $\partial X$; or if some nonempty closed, invariant set in $\partial …
We generalize the theory of gradient flows of semi-convex functions on CAT(0)-spaces, developed by Mayer and Ambrosio--Gigli--Savaré, to CAT(1)-spaces. The key tool is the so-called "commutativity" representing a Riemannian nature of the space, and all results hold true also for metric spaces satisfying the commutativity with semi-convex squared distance functions. Our approach combining the semi-convexity of the squared distance function with a Riemannian property of the space seems to be of independent interest, and can be compared with Savaré's work on the local angle condition under lower …
