An infinitely generated intersection of geometrically finite hyperbolic groups

Two discrete, geometrically finite subgroups of the isometries of hyperbolic n-space () are defined whose intersection is infinitely generated. This settles, in dimensions 4 and above, a long-standing question in Kleinian and hyperbolic groups reiterated at a problem session chaired by Bernard Maskit at the AMS meeting 898, March 3-5, 1995, a conference in honor of Bernard Maskit's 60th birthday.

     

Limit sets of geometrically finite groups acting on Busemann spaces

In this paper, we investigate limit sets of geometrically finite groups acting on Busemann spaces. We show a Busemann space analogue of several results proved by A. Ranjbar-Motlagh for geometrically finite groups acting on hyperbolic spaces in the sense of Gromov.     

Automatic structures,rational growth,and geometrically finite hyperbolic groups

Summary We show that the set of equivalence classes of synchronously automatic structures on a geometrically finite hyperbolic groupG is dense in the product of the sets over all maximal parabolic subgroupsP. The set of equivalence classes of biautomatic structures onG is isomorphic to the product of the sets over the cusps (conjugacy classes of maximal parabolic subgroups) ofG. Each maximal parabolicP is a virtually abelian group, so and were computed in [NS1].We show that any geometrically finite hyperbolic group has a generating set for which the full language of geodesics forG is regular. Moreover, the growth function ofG with respect to this generating set is rational. We also determine which automatic structures on such a group are equivalent to geodesic ones. Not all are, though all biautomatic structures are.Oblatum 14-VI-1993 & 4-I-1994Both authors acknowledge support from the NSF for this research.     

Infinite index subgroups and finiteness properties of intersections of geometrically finite groups

We explore which types of finiteness properties are possible for intersections of geometrically finite groups of isometries in negatively curved symmetric rank one spaces. Our main tool is a twist construction which takes as input a geometrically finite group containing a normal subgroup of infinite index with given finiteness properties and infinite Abelian quotient, and produces a pair of geometrically finite groups whose intersection is isomorphic to the normal subgroup. We produce several examples of such intersections of geometrically finite groups including finitely generated but not finitely presented discrete subgroups.     

Small eigenvalues of geometrically finite manifolds

Let M be a complete geometrically finite manifold of bounded negative curvature, infinite volume, and dimension at least 3.We give both a lower bound for the bottom of the spectrum of M and an upper bound for the number of the small eigenvalues of M. These bounds only depend on the dimension, curvature bounds and the volume of the oneneighborhood of the convex core.     

Smooth classification of geometrically finite one-dimensional maps

The scaling function of a one-dimensional Markov map is defined and studied. We prove that the scaling function of a non-critical geometrically finite one-dimensional map is Hölder continuous, while the scaling function of a critical geometrically finite one-dimensional map is discontinuous. We prove that scaling functions determine Lipschitz conjugacy classes, and moreover, that the scaling function and the exponents and asymmetries of a geometrically finite one-dimensional map are complete -invariants within a mixing topological conjugacy class.

     

Algebraic limits of geometrically finite manifolds are tame

No Abstract. . Received: March 2004 Revision: November 2004 Accepted: June 2005     

Resolvent of the Laplacian on geometrically finite hyperbolic manifolds

For geometrically finite hyperbolic manifolds Γ\ℍ ⁿ⁺¹, we prove the meromorphic extension of the resolvent of Laplacian, Poincaré series, Einsenstein series and scattering operator to the whole complex plane. We also deduce the asymptotics of lattice points of Γ in large balls of ℍ ⁿ⁺¹ in terms of the Hausdorff dimension of the limit set of Γ.     

Radicals and Plotkin's problem concerning geometrically equivalent groups

If and are groups and is a normal subgroup of , then the -closure of in is the normal subgroup of . In particular, is the -radical of . Plotkin calls two groups and geometrically equivalent, written , if for any free group of finite rank and any normal subgroup of the -closure and the -closure of in are the same. Quasi-identities are formulas of the form for any words in a free group. Generally geometrically equivalent groups satisfy the same quasi-identities. Plotkin showed that nilpotent groups and satisfy the same quasi-identities if and only if and are geometrically equivalent. Hence he conjectured that this might hold for any pair of groups. We provide a counterexample.

     

Algorithmically finite groups

We call a group Galgorithmically finite if no algorithm can produce an infinite set of pairwise distinct elements of G. We construct examples of recursively presented infinite algorithmically finite groups and study their properties. For instance, we show that the Equality Problem is decidable in our groups only on strongly (exponentially) negligible sets of inputs.