Thick metric spaces,relative hyperbolicity,and quasi-isometric rigidity

We study the geometry of non-relatively hyperbolic groups. Generalizing a result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup. This implies that a group being relatively hyperbolic with non-relatively hyperbolic peripheral subgroups is a quasi-isometry invariant. As an application, Artin groups are relatively hyperbolic if and only if freely decomposable. We also introduce a new quasi-isometry invariant of metric spaces called metrically thick, which is sufficient for a metric space to be non-hyperbolic relative to any non-trivial collection of subsets. Thick finitely generated groups include: mapping class groups of most surfaces; outer automorphism groups of most free groups; certain Artin groups; and others. Non-uniform lattices in higher rank semisimple Lie groups are thick and hence non-relatively hyperbolic, in contrast with rank one which provided the motivating examples of relatively hyperbolic groups. Mapping class groups are the first examples of non-relatively hyperbolic groups having cut points in any asymptotic cone, resolving several questions of Drutu and Sapir about the structure of relatively hyperbolic groups. Outside of group theory, Teichmüller spaces for surfaces of sufficiently large complexity are thick with respect to the Weil–Peterson metric, in contrast with Brock–Farb’s hyperbolicity result in low complexity.     

Common fixed points of a nonexpansive semigroup and a convergence theorem for Mann iterations in geodesic metric spaces

First, we consider a strongly continuous semigroup of nonexpansive mappings defined on a closed convex subset of a complete CAT(0) space and prove a convergence of a Mann iteration to a common fixed point of the mappings. This result is motivated by a result of Kirk (2002) and of Suzuki (2002). Second, we obtain a result on limits of subsequences of Mann iterations of multivalued nonexpansive mappings on metric spaces of hyperbolic type, which leads to a convergence theorem for nonexpansive mappings on these spaces.     

A Combination Theorem for Metric Bundles

We introduce the notion of metric (graph) bundles which provide a coarse-geometric generalization of the notion of trees of metric spaces a la Bestvina?CFeighn in the special case that the inclusions of the edge spaces into the vertex spaces are uniform coarsely surjective quasi-isometries. We prove the existence of quasi-isometric sections in this generality. Then we prove a combination theorem for metric (graph) bundles that establishes sufficient conditions, particularly flaring, under which the metric bundles are hyperbolic. We use this to give examples of surface bundles over hyperbolic disks, whose universal cover is Gromov-hyperbolic. We also show that in typical situations, flaring is also a necessary condition.     

Peripheral splittings of groups

We define the notion of a ``peripheral splitting' of a group. This is essentially a representation of the group as the fundamental group of a bipartite graph of groups, where all the vertex groups of one colour are held fixed--the ``peripheral subgroups'. We develop the theory of such splittings and prove an accessibility result. The theory mainly applies to relatively hyperbolic groups with connected boundary, where the peripheral subgroups are precisely the maximal parabolic subgroups. We show that if such a group admits a non-trivial peripheral splitting, then its boundary has a global cut point. Moreover, the non-peripheral vertex groups of such a splitting are themselves relatively hyperbolic. These results, together with results from elsewhere, show that under modest constraints on the peripheral subgroups, the boundary of a relatively hyperbolic group is locally connected if it is connected. In retrospect, one further deduces that the set of global cut points in such a boundary has a simplicial treelike structure.

     

Connectedness properties of limit sets

We study convergence group actions on continua, and give a criterion which ensures that every global cut point is a parabolic fixed point. We apply this result to the case of boundaries of relatively hyperbolic groups, and consider implications for connectedness properties of such spaces.

     

Quasi-isometries between visual hyperbolic spaces

We prove that a power quasi-symmetric (or PQ-symmetric) homeomorphism between two complete metric spaces can be extended to a quasi-isometry between their hyperbolic approximations. This result can be used to prove that two visual Gromov hyperbolic spaces are quasi-isometric if and only if there is a PQ-symmetric homeomorphism between their boundaries with bounded visual metrics. Also, in the case of trees, we prove that two geodesically complete trees are quasi-isometric if and only if there is a PQ-symmetric homeomorphism between their boundaries with visual metrics based at infinity. We also give a characterization for a map to be PQ-symmetric based on the relative distortion of subsets.     

Hyperbolic groups have finite asymptotic dimension

We detail a proof of a result of Gromov, that hyperbolic groups (and metric spaces) have finite asymptotic dimension. This fact has become important in recent work on the Novikov conjecture.

     

Altitude properties and characterizations of inner product spaces

New characterizations of real inner product spaces (euclidean spaces) among metric spaces are obtained from familiar formulas expressing the altitude (height) of a triangle as a function of the lengths of its sides. Other properties related to the altitude of a triangle are also shown to result in characterizations of euclidean spaces, or euclidean and hyperbolic spaces.     

Stability of transonic shock for supersonic flow past a wedge

When steady supersonic flow hits a slim wedge, there may appear an oblique transonic shock attached to the vertex of the wedge, if the downstream pressure is rather large. This paper studies stability in certain weighted partial Hölder spaces of the oblique transonic shock attached to the vertex of a wedge, which is against steady supersonic flows, under perturbations of the upstream flow and the profile of the wedge. We show that under reasonable conditions on the upcoming supersonic flow and the slope of the wedge, such transonic shocks are structural stable. Mathematically, we solve an elliptic–hyperbolic mixed type in an unbounded domain, and the flow field is proved to be C¹. Copyright © 2015 John Wiley & Sons, Ltd.     

On Fourier coefficients of Maass cusp forms in 3-dimensional hyperbolic space

In this article we establish the analogue of a theorem of Kuznetsov (theorem 6 of [3]) in the case of 3-dimensional hyperbolic space. We also consider a generalization of this result for higher dimensional hyperbolic spaces and discuss the relevant ingredients of a proof. Dedicated to the memory of Professor K G Ramanathan     

Separately normal maps

It is shown that a separately normal map is holomorphic and a separately normal family which is separately uniformly normal is a normal family extending a result by Barth that a separately holomorphic map into hyperbolic spaces is holomorphic and a separately normal family of maps into hyperbolic spaces is a normal family.     

Groups acting on tree-graded spaces and splittings of relatively hyperbolic groups

Tree-graded spaces are generalizations of R-trees. They appear as asymptotic cones of groups (when the cones have cut-points). Since many questions about endomorphisms and automorphisms of groups, solving equations over groups, studying embeddings of a group into another group, etc. lead to actions of groups on the asymptotic cones, it is natural to consider actions of groups on tree-graded spaces. We develop a theory of such actions which generalizes the well-known theory of groups acting on R-trees. As applications of our theory, we describe, in particular, relatively hyperbolic groups with infinite groups of outer automorphisms, and co-Hopfian relatively hyperbolic groups.     

Variations on a Proof of a Borderline Bourgain-Brezis Sobolev Embedding Theorem

This paper offers a variant of a proof of a borderline Bourgain-Brezis Sobolev embedding theorem on R~n. The authors use this idea to extend the result to real hyperbolic spaces H~n.     

Coronae of product spaces and the coarse Baum–Connes conjecture

We study the coarse Baum–Connes conjecture for product spaces and product groups. We show that a product of CAT(0) groups, polycyclic groups and relatively hyperbolic groups which satisfy some assumptions on peripheral subgroups, satisfies the coarse Baum–Connes conjecture. For this purpose, we construct and analyze an appropriate compactification and its boundary, “corona”, of a product of proper metric spaces.     

Operator norm localization property of relative hyperbolic group and graph of groups

In this article we study the spaces which have operator norm localization property. We prove that a finitely generated group Γ which is strongly hyperbolic with respect to a collection of finitely generated subgroups {H₁,…,Hn} has operator norm localization property if and only if each Hi, i=1,2,…,n, has operator norm localization property. Furthermore we prove the following result. Let π be the fundamental group of a connected finite graph of groups with finitely generated vertex groups GP. If GP has operator norm localization property for all vertices P then π has operator norm localization property.     

The isomorphism problem for toral relatively hyperbolic groups

We provide a solution to the isomorphism problem for torsion-free relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsion-free hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually free groups (Bumagin, Kharlampovich and Miasnikov [14]). We also give a solution to the homeomorphism problem for finite volume hyperbolic n-manifolds, for n≥3. In the course of the proof of the main result, we prove that a particular JSJ decomposition of a freely indecomposable torsion-free relatively hyperbolic group with abelian parabolics is algorithmically constructible.     

Hyperbolic equations, function spaces with exponential weights and Nemytskij operators

Different problems in the theory of hyperbolic equations bases on function spaces of Gevrey type are studied. Beside the original Gevrey classes, spaces defined by the behaviour of the Fourier transform were also used to prove basic results about the well-posedness of Cauchy problems for non-linear hyperbolic systems. In these approaches only the algebra property of the function spaces was used to include analytic non-linearities. Here we will generalize this dependence. First we investigate superposition operators in spaces with exponential weights. Then we show in concrete situations how a priori estimates of strictly hyperbolic type lead to the well-posedness of certain semi-linear hyperbolic Cauchy problems in suitable function spaces with exponential weights of Gevrey type. Mathematics Subject Classification (2000) 46E35, 35L80, 35L15, 47H30     

Strong Embeddability for Groups Acting on Metric Spaces

The strong embeddability is a notion of metric geometry, which is an intermediate property lying between coarse embeddability and property A. In this paper, the permanence properties of strong embeddability for groups acting on metric spaces are studied. The authors show that a finitely generated group acting on a finitely asymptotic dimension metric space by isometries whose $K$-stabilizers are strongly embeddable is strongly embeddable. Moreover, they prove that the fundamental group of a graph of groups with strongly embeddable vertex groups is also strongly embeddable.     

Unitarizable representations and fixed points of groups of biholomorphic transformations of operator balls

We show that the open unit ball of the space of operators from a finite-dimensional Hilbert space into a separable Hilbert space (we call it “operator ball”) has a restricted form of normal structure if we endow it with a hyperbolic metric (which is an analogue of the standard hyperbolic metric on the unit disc in the complex plane). We use this result to get a fixed point theorem for groups of biholomorphic automorphisms of the operator ball. The fixed point theorem is used to show that a bounded representation in a separable Hilbert space which has an invariant indefinite quadratic form with finitely many negative squares is unitarizable (equivalent to a unitary representation). We apply this result to find dual pairs of invariant subspaces in Pontryagin spaces. In Appendix A we present results of Itai Shafrir about hyperbolic metrics on the operator ball.