Quasi-hyperbolic planes in hyperbolic groups

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

     

Height in splittings of hyperbolic groups

SupposeH is a hyperbolic subgroup of a hyperbolic groupG. Assume there existsn > 0 such that the intersection ofn essentially distinct conjugates ofH is always finite. Further assumeG splits overH with hyperbolic vertex and edge groups and the two inclusions ofH are quasi-isometric embeddings. ThenH is quasiconvex inG. This answers a question of Swarup and provides a partial converse to the main theorem of [23].     

A topological space for which graph embeddability is undecidable

There exists a path-connected subspace of the plane for which graph embeddability is undecidable.     

Dehn filling in relatively hyperbolic groups

We introduce a number of new tools for the study of relatively hyperbolic groups. First, given a relatively hyperbolic group G, we construct a nice combinatorial Gromov hyperbolic model space acted on properly by G, which reflects the relative hyperbolicity of G in many natural ways. Second, we construct two useful bicombings on this space. The first of these, preferred paths, is combinatorial in nature and allows us to define the second, a relatively hyperbolic version of a construction of Mineyev. As an application, we prove a group-theoretic analog of the Gromov-Thurston 2π Theorem in the context of relatively hyperbolic groups. The first author was supported in part by NSF Grant DMS-0504251. The second author was supported in part by an NSF Mathematical Sciences Post-doctoral Research Fellowship. Both authors thank the NSF for their support. Most of this work was done while both authors were Taussky-Todd Fellows at Caltech.     

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.

     

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.     

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.     

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     

Higher dimensional isoperimetric functions in hyperbolic groups

We introduce the notion of an -combing and use it to show that hyperbolic groups satisfy linear isoperimetric inequalities for filling real cycles in each positive dimension. S. Gersten suggested the concept of metabolicity (over or ) for groups which implies hyperbolicity. Metabolicity admits several equivalent definitions: by vanishing of -cohomology, using combings, and others. We prove several criteria for a group to be hyperbolic, -metabolicity being among them. In particular, a finitely presented group G is hyperbolic iff for any normed vector space V and any . Received December 9, 1998     

Structure and rigidity in hyperbolic groups I

We introduce certain classes of hyperbolic groups according to their possible actions on real trees. Using these classes and results from the theory of (small) group actions on real trees, we study the structure of hyperbolic groups and their automorphism group.The second author was partially supported by an NSF grant.     

Height in splittings of relatively hyperbolic groups

Geometriae Dedicata - Given a finite graph of relatively hyperbolic groups with its fundamental group relatively hyperbolic and edge groups quasi-isometrically embedded and relatively quasiconvex...     

Canonical representatives and equations in hyperbolic groups

Summary We use canonical representatives in hyperbolic groups to reduce the theory of equations in (torsion-free) hyperbolic groups to the theory in free groups. As a result we get an effective procedure to decide if a system of equations in such groups has a solution. For free groups, this question was solved by Makanin [Ma]|and Razborov [Ra]. The case of quadratic equations in hyperbolic groups has already been solved by Lysenok [Ly]. Our whole construction plays an essential role in the solution of the isomorphism problem for (torsion-free) hyperbolic groups ([Se1],[Se2]).Oblatum 1-1992 & 1-XI-1994Partially supported by NSF grant DMS-9305848     

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.