# Finite groups and hyperbolic manifolds

@article{Belolipetsky2005FiniteGA, title={Finite groups and hyperbolic manifolds}, author={Mikhail V. Belolipetsky and Alexander Lubotzky}, journal={Inventiones mathematicae}, year={2005}, volume={162}, pages={459-472} }

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.

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## References

SHOWING 1-10 OF 25 REFERENCES

### Free quotients and the first betti number of some hyperbolic manifolds

- Mathematics
- 1996

In this note we present a very simple method of proving that some hyperbolic manifoldsM have finite sheeted covers with positive first Betti number. The method applies to the standard arithmetic…

### Strong approximation for Zariski-dense subgroups of semi-simple algebraic groups

- Mathematics
- 1984

(1.1) THEOREM. Let k be an algebraically closed field of characteristic different from 2 and 3, and G an almost simple, connected and simply connected algebraic group defined over k. Let F be a…

### On asymmetric hyperbolic manifolds

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 2005

We show that for every $n\,{\geq}\,2$ there exists closed hyperbolic n-manifolds for which the full group of orientation preserving isometries is trivial.

### Strong approximation for Zariski dense subgroups over arbitrary global fields

- Mathematics
- 2000

Abstract. Consider a finitely generated Zariski dense subgroup
$ \Gamma $ of a connected simple algebraic group G over a global field F. An important aspect of strong approximation is the question…

### Discrete subgroups of Lie groups

- Mathematics
- 1972

Preliminaries.- I. Generalities on Lattices.- II. Lattices in Nilpotent Lie Groups.- III. Lattices in Solvable Lie Groups.- IV. Polycyclic Groups and Arithmeticity of Lattices in Solvable Lie…

### RINGS OF DEFINITION OF DENSE SUBGROUPS OF SEMISIMPLE LINEAR GROUPS

- Mathematics
- 1971

We investigate the question: What is the smallest ring over which the elements of a dense subgroup (in the Zariski topology) of a semisimple algebraic group can be written down simultaneously for…

### Discrete Subgroups of Semisimple Lie Groups

- Mathematics
- 1991

1. Statement of Main Results.- 2. Synopsis of the Chapters.- 3. Remarks on the Structure of the Book, References and Notation.- 1. Preliminaries.- 0. Notation, Terminology and Some Basic Facts.- 1.…

### Graphs of Degree Three with a Given Abstract Group

- MathematicsCanadian Journal of Mathematics
- 1949

1. Introduction. In his well-known book on graphs [1] König proposed the following problem: “When can a given abstract group be represented as the group of the automorphisms of a (finite) graph, and…

### Enumerating Boundedly Generated Finite Groups

- Mathematics
- 2001

Abstract We prove a conjecture of Mann and Pyber which estimates the number of finite groups of a given order and a given number of generators. This implies that the normal subgroup growth of free…