Cubulating small cancellation groups

We study the B(6) and B(4)-T(4) small cancellation groups. These classes include the usual C(1/6) and C(1/4)-T(4) metric small cancellation groups. We show that every finitely presented B(4)-T(4) or word-hyperbolic B(6) group acts properly discontinuously and cocompactly on a CAT(0) cube complex. We show that finitely generated infinite B(6) and B(4)-T(4) groups have codimension 1 subgroups and thus do not have property (T). We show that a finitely presented B(6) group is wordhyperbolic if and only if it contains no subgroup.     

Cubulating random groups in the square model

Our main result is that for densities < 3/10 a random group in the square model has the Haagerup property and is residually finite. Moreover, we generalize the Isoperimetric Inequality to some class of non-planar diagrams and, using this, we introduce a system of modified hypergraphs providing the structure of a space with walls on the Cayley complex of a random group. Then we show that the natural action of a random group on this space with walls is proper, which gives the proper action of a random group on a CAT(0) cube complex.     

On the hyperbolic limit points of groups acting on hyperbolic spaces

We study the hyperbolic limit points of a groupG acting on a hyperbolic metric space, and consider the question of whether any attractive limit point corresponds to a unique repulsive limit point. In the special case whereG is a (non-elementary) finitely generated hyperbolic group acting on its Cayley graph, the answer is affirmative, and the resulting mapg ⁺↦g ⁻, is discontinuous everywhere on the hyperbolic boundary. We also provide a direct, combinatorial proof in the special case whereG is a (non-abelian) free group of finite type, by characterizing algebraically the hyperbolic ends ofG. Partially supported by a grant from M.U.R.S.T., Italy.     

Growth of relatively hyperbolic groups

We show that a finitely generated group that is hyperbolic relative to a collection of proper subgroups either is virtually cyclic or has uniform exponential growth.

     

On the structure of two-generated hyperbolic groups

We show that every two-generated torsion-free one-ended word-hyperbolic group has virtually cyclic outer automorphism group. This is done by computing the JSJ-decomposition for two-generator hyperbolic groups. We further prove that two-generated torsion-free word-hyperbolic groups are strongly accessible. This means that they can be constructed from groups with no nontrivial cyclic splittings by applying finitely many free products with amalgamation and HNN-extensions over cyclic subgroups. Received June 25, 1998     

Rings associated with hyperbolic groups

Abstract We define quasiconvexity cone Qcone(τ) over an infinite hyperbolic (in the sense of Gromov) group τ as the set of conjugacy classes of infinite quasiconvex subgroups H?τ and show that the abelian group of Qcone(τ)-divisors, i.e. finite sums of points from Qcone(τ) with integer coefficients, can be equipped with a natural structure of commutative associative ring with identity. Euler characteristic can be considered as a rational-valued function on Qcone(τ). This approach gives another point of view on the strengthened form of Hanna Neumann's conjecture on the maximal rank of the intersection of two finitely generated subgroups of the free group on two generators.     

On groups of hyperbolic length

We investigate the recently introduced notion of rotation numbers for periodic orbits of interval maps. We identify twist orbits, that is those orbits that are the simplest ones with given rotation number. We estimate from below the topological entropy of a map having an orbit with given rotation number. Our estimates are sharp: there are unimodal maps where the equality holds. We also discuss what happens for maps with larger modality. In the Appendix we present a new approach to the problem of monotonicity of entropy in one-parameter families of unimodal maps. This work was partially done during the first author’s visit to IUPUI (funded by a Faculty Research Grant from UAB Graduate School) and his visit to MSRI (the research at MSRI funded in part by NSF grant DMS-9022140) whose support the first author acknowledges with gratitude. The second author was partially supported by NSF grant DMS-9305899, and his gratitude is as great as that of the first author.     

Quasi-hyperbolic planes in hyperbolic groups

The hyperbolic plane admits a quasi-isometric embedding into every hyperbolic group which is not virtually free.

     

Finite subgroups of hyperbolic groups

Let G be a hyperbolic group which is -thin w.r.t. a finite generating set X. We show that every finite subgroup of G is conjugate to a subgroup each element of which has length at most 2 + 1relative to X. Translated fromAlgebra i Logika, Vol. 34, No. 6, pp. 619-622, November-December, 1995.Supported by RFFR grant No. 93-11-1501 and by ISF grant RPC000.Supported by RFFR grant No. 93-011-16171.     

Coxeter groups and hyperbolic manifolds

The theory of Coxeter groups is used to provide an algebraic construction of finite volume hyperbolic manifolds. Combinatorial properties of finite images of these groups can be used to compute the volumes of the resulting manifolds. Three examples, in 4,5 and 6-dimensions, are given, each of very small volume, and in one case of smallest possible volume.The author is grateful to Patrick Dorey for a number of helpful conversations.Revised version: 22 December 2003     

Finite groups and hyperbolic manifolds

The isometry group of a compact n-dimensional hyperbolic manifold is known to be finite. We show that for every n≥2, every finite group is realized as the full isometry group of some compact hyperbolic n-manifold. The cases n=2 and n=3 have been proven by Greenberg (1974) and Kojima (1988), respectively. Our proof is non constructive: it uses counting results from subgroup growth theory to show that such manifolds exist.     

On the definition of word hyperbolic groups

Formal languages based on multiplication tables of finitely generated groups are investigated and used to give a linguistic characterization of word hyperbolic groups. Received: 19 October 2000 / in final version: 6 June 2001 / Published online: 1 February 2002     

The automorphism groups of the vertex operator algebras V + L : general case

In this article, we give a method of calculating the automorphism groups of the vertex operator algebras V⁺_L associated with even lattices L. For example, by using this method we determine the automorphism groups of V⁺_L for even lattices of rank one, two and three, and even unimodular lattices. The author was supported by the Japan Society for the Promotion of Science Research Fellowships for Young Scientists and COE grant of Hokkaido University.     

Subcoercivity and subelliptic operators on Lie groups II: The general case

Let (,G, U) be a continuous representation of a Lie groupG by bounded operatorsg U (g) on the Banach space and let (, ,dU) denote the representation of the Lie algebra obtained by differentiation. Ifa ₁, ...,a _d is a Lie algebra basis of ,A _i =dU (a _i ) and whenever =(i ₁, ...,i _k ) we reconsider the operators

Limit sets of relatively hyperbolic groups

In this paper, we prove a limit set intersection theorem in relatively hyperbolic groups. Our approach is based on a study of dynamical quasiconvexity of relatively quasiconvex subgroups. Using dynamical quasiconvexity, many well-known results on limit sets of geometrically finite Kleinian groups are derived in general convergence groups. We also establish dynamical quasiconvexity of undistorted subgroups in finitely generated groups with nontrivial Floyd boundaries.     

Complete growth functions of hyperbolic groups

. We study a generalization of the growth functions of finitely generated groups, namely the growth functions Σ _{g ∈ G} gz ^{| g |} with coefficients in the group ring ℤ[G]. Rationality and methods of computation of such functions are discussed, in particular for hyperbolic groups. The complete growth functions of surface groups are explicitly computed. The operator and geodesic growth functions are also studied. Oblatum 20-IX-1996 & 13-I-1997     

Strongly geodesically automatic groups are hyperbolic

Oblatum VII-1994 & 1-III-1995     

Two-transitive groups on a hyperbolic unital

A classification given previously of all projective translation planes of order q² that admit a collineation group G admitting a two-transitive orbit of q+1 points is applied to show that the only projective translation planes of order q² admitting a hyperbolic unital acting two-transitively on a secant are the Desarguesian planes and the unital is a Buekenhout hyperbolic unital.