# Linear Representations of the Automorphism Group of a Free Group

@article{Grunewald2006LinearRO, title={Linear Representations of the Automorphism Group of a Free Group}, author={Fritz Grunewald and Alexander Lubotzky}, journal={Geometric and Functional Analysis}, year={2006}, volume={18}, pages={1564-1608} }

Abstract.Let Fn be the free group on n ≥ 2 elements and Aut(Fn) its group of automorphisms. In this paper we present a rich collection of linear representations of Aut(Fn) arising through the action of finite-index subgroups of it on relation modules of finite quotient groups of Fn. We show (under certain conditions) that the images of our representations are arithmetic groups.

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## References

SHOWING 1-10 OF 32 REFERENCES

Finite Quotients of the Automorphism Group of a Free Group

- MathematicsCanadian Journal of Mathematics
- 1977

Let G and F be groups. A G-defining subgroup of F is a normal subgroup N of F such that F/N is isomorphic to G. The automorphism group Aut (F) acts on the set of G-defining subgroups of F. If G is…

Low-dimensional Linear Representations of Aut F N ; N 3

- Mathematics

We classify all complex representations of Aut F n ; the au-tomorphism group of the free group F n (n 3); of dimension 2n ? 2: Among those representations is a new representation of dimension n +1…

On the Spread of Finite Simple
Groups

- MathematicsComb.
- 2003

An asymptotic result is proved characterizing the finite simple groups of bounded spread and it is shown that it is at least 2, with possibly finitely many exceptions.

ON THE GROUP SL2 OVER DEDEKIND RINGS OF ARITHMETIC TYPE

- Mathematics
- 1972

It is proved that the group of matrices of order two with determinant 1 over a Dedekind ring of arithmetic type is generated by elementary matrices if there are infinitely many invertible elements in…

Linear representations of finite groups

- MathematicsGraduate texts in mathematics
- 1977

Representations and characters: generalities on linear representations character theory subgroups, products, induced representation compact groups examples. Representations in characteristic zero:…

Low-dimensional representations of Aut (F2)

- Mathematics
- 1996

LetF2 be the free group of rank two, and Φ2 its automorphism group. We consider the problem of describing the representations of Φ2 of degreen for small values ofn. Our main result is the…

On systems of generators of arithmetic subgroups of higher rank groups

- Mathematics
- 1994

We show that any two maximal disjoint unipotent subgroups of an irreducible non-cocompact lattice in a Lie group of rank atleast two generates a lattice. The proof uses techniques of the solution of…

The Proalgebraic Completion of Rigid Groups

- Mathematics
- 2002

A finitely generated group Γ is called representation rigid (briefly, rigid) if for every n, Γ has only finitely many classes of simple ℂ representations in dimension n. Examples include higher rank…

Groups Acting on Hyperbolic Space: Harmonic Analysis and Number Theory

- Mathematics
- 1997

1. Three-Dimensional Hyperbolic Space.- 2. Groups Acting Discontinuously on Three-Dimensional Hyperbolic Space.- 3. Automorphic Functions.- 4. Spectral Theory of the Laplace Operator.- 5. Spectral…