論文

We show that finitely generated groups which are Liouville and without infinite finite-dimensional linear representations must have a global fixed point whenever they act by isometry on a finite-dimensional complete CAT(0)-space. This provides a partial answer to an old question in geometric group theory and proves partly a conjecture formulated by Norin, Osajda, and Przytycki. It applies in particular to Grigorchuk's groups of intermediate growth and other branch groups as well as to simple groups with the Liouville property such as those found by Matte Bon and by Nekrashevych. The method of proof uses ultralimits, equivariant harmonic maps, subharmonic functions, horofunctions and random walks.

In this paper we use the combinatorial harmonic map theory to study the isometric actions of discrete groups on Hadamard spaces. Given a finitely generated group acting by automorphisms, properly discontinuously and cofinitely on a simplicial complex and its isometric action on a Hadamard spaces, we formulate criterions for the action to have a global fixed point.

We show a fixed-point property of certain random groups for a wide class of CAT(0) spaces. The model of random groups under consideration is given as the set of presentations (S, R), where S is a generating set and the set of relations R is a subset of the set of all plain words of the same length with suitably fixed density. Our main theorem says that, with high probability, groups obtained by such presentations have the fixed-point property for all CAT(0) spaces having bounded singularities.

We define the Teichmüller pseudodistance on spaces of flat conformal structures by the same manner as classical Teichmüller distance on the Teichmüller space of Riemann surfaces. We will prove that for compact manifolds this pseudodistance becomes a complete distance. We will also prove similar results for noncompact manifolds under certain assumptions. © 1998 American Mathematical Society.

We prove that a random group of the graph model associated with a sequence of expanders has the fixed-point property for a certain class of CAT (0) spaces. We use Gromov's criterion for the fixed-point property in terms of the growth of n-step energy of equivariant maps from a finitely generated group into a CAT.(0) space, for which we give a detailed proof. We estimate a relevant geometric invariant of the tangent cones of the Euclidean buildings associated with the groups PGL (m, Q(r)), and deduce from the general result above that the same random group has the fixed-point property for all of these Euclidean buildings with m bounded from above.

We prove a certain gap theorem concerning the Yamabe invariants for compact conformally flat 4-manifolds with positive Euler numbers.

We consider deformations of flat conformal structures from a view-point of connected sum decomposition of conformally flat manifolds.

We prove that finitely generated amenable groups acting on CAT(0) spaces satisfy the following alternative: either every action on a geodesically complete CAT(0) space with bounded geometry (or finite dimension) has a global fixed point, or the group admits a fixed-point-free action on $\mathbb{R}^n$. As a consequence, finitely generated amenable torsion groups and finitely generated virtually simple amenable groups cannot act nontrivially on geodesically complete CAT(0) spaces with bounded geometry or on finite-dimensional complete CAT(0) spaces. The proof relies on a Kazhdan-type rigidity theorem for groups with the Euclidean fixed point property: if such a group acts on a geodesically complete CAT(0) space of bounded geometry with almost fixed points, then it has a genuine fixed point. This yields several further corollaries, including a rigidity dichotomy for drift and that any finitely generated torsion group acting on a geodesically complete visibility CAT(0) space with bounded geometry must have a global fixed point. These results make substantial progress on the longstanding problem of understanding actions of torsion groups on CAT(0) spaces.

We give a suffcient condition for a higher dimensional Kleinian group Gamma subset of Isom(Hn+1) to be convex cocompact in terms of the critical exponent of. As a consequence, we see that the fundamental group of a compact conformally at manifold with positive scalar curvature is hyperbolic in the sense of Gromov. We give some other applications to geometry and topology of conformally at manifolds with positive scalar curvature.

We show the quasiconformal stability for torsion-free convex cocompact Kleinian groups acting on higher dimensional hyperbolic spaces. As an application, we prove an embedding theorem of a space of at conformal structures on a certain class of compact manifolds. © 1999 American Mathematical Society.