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.     

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     

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.     

On the structure of hyperbolic manifolds

Forn≥2, we quantify the Margulis constant ε(n) giving rise to a thick and thin decomposition of hyperbolicn-manifolds of finite volume. As a consequence, we obtain new universal lower bounds for the volume and Gromov's invariant as well as a geometrical inequality between injectivity radius and diameter for compact manifolds. Finally, we concretise the upper bound for the counting function of hyperbolic manifolds of dimension >4 as described by Burger, Gelander, Lubotzky and Mozes. Partially supported by Schweizerischer Nationalfonds No. 20-61379.00 and 20-67619.02.     

On two-generated modules over valuation domains

Archiv der Mathematik -     

On the Selberg trace formula for strictly hyperbolic groups

We show that, for the case of strictly hyperbolic groups, the right-hand side of the Selberg trace formula admits a representation in the form of a series in the eigenvalues of the Laplacian. The behavior of the Minakshisundaram function as t → 0 and t → ∞ is studied. Countably many conditions satisfied by the spectrum of the Laplacian are obtained in explicit form.     

Endomorphisms of two-generated metabelian groups that induce the identity modulo the derived subgroup

Partially supported by MPI, Italy. Authors are members of CNR-GNSAGA, Italy.     

On quasiconvexity and relatively hyperbolic structures on groups

Let G be a group which is hyperbolic relative to a collection of subgroups H₁{\mathcal{H}_1}, and it is also hyperbolic relative to a collection of subgroups H₂{\mathcal{H}_2}. Suppose that H₁ ì H₂{\mathcal{H}_1 \subset \mathcal{H}_2}. We characterize when a relative quasiconvex subgroup of (G, H₂){(G, \mathcal H_2)} is still relatively quasiconvex in (G, H₁){(G, \mathcal H_1)}. We also show that relative quasiconvexity is preserved when passing from (G, H₁){(G, \mathcal H_1)} to (G, H₂){(G, \mathcal H_2)}. Applications are discussed.     

On the structure of submanifolds in the hyperbolic space

Monatshefte für Mathematik - Let $$M^n$$ be a complete submanifold in the hyperbolic space $$mathbb {H}^{n+m}$$ . We show the vanishing of the Betti numbers $$beta _p(M)$$ , $$1 le p le...     

The structure of 3-manifolds with two-generated fundamental group

We provide a structure theorem for 3-manifolds with 2-generated fundamental group and non-trivial JSJ-decomposition. We further give a number of applications.     

On the residual finiteness of outer automorphisms of relatively hyperbolic groups

We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable relatively hyperbolic group is residually finite. As a direct consequence, we obtain that the outer automorphism group of a limit group is residually finite.     

On the structure of Selmer groups

This paper was partly prepared during my stay at the Max-Planck-Institut für Mathematik in Bonn. I want to express my gratitude for the support and the hospitality provided by this institute     

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 ring of global sections of the blowing-up along a two-generated ideal

The effective relations and the relation type of an ideal generated by two regular elements can be expressed in terms of the ring of global sections of the blowing-up along the ideal. As a consequence, we find the minimal overring where the ideal is syzygetic or of linear type.     

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.     

ON THE STRUCTURE OF THEATTRACTORS OF HYPERBOLICITERATED FUNCTION SYSTEM

1.IntroductionThefractalsgeneratedbytheattractorsofiteratedfunctionsystems(i.f.s.)havebeenresearchedbymanyauthorsfl--3'6'7'9].ByaniteratedfunctionsystemwemeanacompactmetricspaceXtogetherwithacollectionofcontinuousmapsTI,T2,'tTNonit,denotedby(X,TI,',TN).IfalltheTi'sarecontractionswecall(X;TI,',TN)ahyperboliciteratedfunctionsystem(h.i.f.s.).NForanh.i.f.s.thereexistsacompactsubsetAofX,suchthatA=.UTi(A).Aiscalledtheattractoroftheh.i.f.s.DenoteZ=(1,2,',N)N,anddefineametricdonZby…