# 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}
}
• Published 29 June 2004
• Mathematics
• Inventiones mathematicae
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.

### Countable groups are mapping class groups of hyperbolic 3-manifolds

• Mathematics
• 2005
We prove that for every countable group G there exists a hyperbolic 3-manifold M such that the isometry group of M, the mapping class group of M, and the outer automorphism group of the fundamental

### Symmetries of exotic smoothings of aspherical space forms

• Mathematics
• 2021
We study finite group actions on smooth manifolds of the form W#Σ, where Σ is an exotic n-sphere and W is either a hyperbolic or a flat manifold. We classify the finite cyclic groups that act freely

### Constructing isospectral manifolds

In this article we construct nonisometric, isospectral manifolds modelled on semisimple Lie groups with finite center and no compact factors. Specifically, our two main results are the construction

### Symmetries of hyperbolic 4-manifolds

• Mathematics
• 2014
In this paper, for each finite group $G$, we construct explicitly a non-compact complete finite-volume arithmetic hyperbolic $4$-manifold $M$ such that $\mathrm{Isom}\,M \cong G$, or

### Aspherical Manifolds with Hyperbolic Fundamental Groups Can ’ t Collapse

We show that a sequence of n-dimensional spaces diffeomorphic to closed aspherical manifolds with non-elementary hyperbolic fundamental groups cannot converge in the Gromov–Hausdorff sense to a lower

### On Symmetry of Flat Manifolds

It is got that every symmetric group can be realized as an outer automorphism group of some Bieberbach group.

### Isospectral locally symmetric manifolds

In this article we construct closed, isospectral, non-isom etric locally symmetric manifolds. We have three main results. First, we construct arbitrarily large sets of closed, isospectral,

### Finiteness properties for some rational Poincaré duality groups

A combination of Bestvina–Brady Morse theory and an acyclic reflection group trick produces a torsion-free finitely presented Q-Poincare duality group which is not the fundamental group of an

### Homotopy groups of the moduli space of metrics of positive scalar curvature

• Mathematics
• 2010
We show by explicit examples that in many degrees in a stable range the homotopy groups of the moduli spaces of Riemannian metrics of positive scalar curvature on closed smooth manifolds can be

## References

SHOWING 1-10 OF 25 REFERENCES

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

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

(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

• Mathematics
Mathematical 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

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

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

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

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.

• R. Frucht
• Mathematics