# Kleinian groups and the rank problem

@article{Kapovich2005KleinianGA, title={Kleinian groups and the rank problem}, author={Ilya Kapovich and Richard Weidmann}, journal={Geometry \& Topology}, year={2005}, volume={9}, pages={375-402} }

We prove that the rank problem is decidable in the class of torsion-free wordhyperbolic Kleinian groups. We also show that every group in this class has only finitely many Nielsen equivalence classes of generating sets of a given cardinality.

#### 14 Citations

On the rank of quotients of hyperbolic groups

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We show that the rank does not decrease if one passes from a torsionfree locally quasi-convex hyperbolic group to the quotient by the normal closure of certain high powered element. An argument… Expand

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We determine the rank of the fundamental group of those hyperbolic 3‐manifolds fibering over the circle whose monodromy is a sufficiently high power of a pseudoAnosov map. Moreover, we show that any… Expand

Rank and Nielsen equivalence in hyperbolic extensions

- Computer Science, Mathematics
- Int. J. Algebra Comput.
- 2019

A theorem of Juan Souto on rank and Nielsen equivalence in the fundamental group of a hyperbolic fibered 3-manifold is generalized to a large class of hyperbols of surfaces groups and free groups by convex cocompact subgroups of Out$(F_n)$. Expand

Generating Tuples of Virtually Free Groups

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We give a complete description of all generating tuples of a virtually free group, i.e., we give a parametrization of Epi(Fn, Г) where n ∈ N and G is a virtually free group.

Generating pairs of 2-bridge knot groups

- Mathematics
- 2009

AbstractWe study Nielsen equivalence classes of generating pairs of Kleinian groups and HNN-extensions. We establish the following facts:
(1)Hyperbolic 2-bridge knot groups have infinitely many… Expand

2 0 D ec 2 01 8 Rank and Nielsen equivalence in hyperbolic extensions

- 2018

In this note, we generalize a theorem of Juan Souto on rank and Nielsen equivalence in the fundamental group of a hyperbolic fibered 3–manifold to a large class of hyperbolic group extensions. This… Expand

RANK OF MAPPING TORI AND COMPANION MATRICES by Gilbert LEVITT and Vassilis METAFTSIS

- 2012

Given an element GL(d Z) , consider the mapping torus defined as the semidirect product G Z Z . We show that one can decide whether G has rank 2 or not (i.e. whether G may be generated by two… Expand

Geometry, Heegaard splittings and rank of the fundamental group of hyperbolic 3-manifolds

- Mathematics
- 2007

In this survey we discuss how geometric methods can be used to study topological properties of 3‐manifolds such as their Heegaard genus or the rank of their fundamental group. On the other hand, we… Expand

Rank of mapping tori and companion matrices

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- 2010

Given $f$ in $GL(d,Z)$, it is decidable whether its mapping torus (the semi-direct product of $Z^d$ with $Z$) may be generated by two elements or not; if so, one can classify generating pairs up to… Expand

Nielsen equivalence in a class of random groups

- Mathematics
- 2016

We show that for every $n\ge 2$ there exists a torsion-free one-ended word-hyperbolic group $G$ of rank $n$ admitting generating $n$-tuples $(a_1,\ldots ,a_n)$ and $(b_1,\ldots ,b_n)$ such that the… Expand

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We determine the rank of the fundamental group of those hyperbolic 3‐manifolds fibering over the circle whose monodromy is a sufficiently high power of a pseudoAnosov map. Moreover, we show that any… Expand

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