Fritz Grunewald died unexpectedly this week, just before his 61st birthday. I never met him but have always been an admirer of his work, and I’d been meaning to post about his lovely paper with Lubotzky, “Linear representations of the automorphism group of the free group.” I’m sorry it takes such a sad event to spur me to get around to this, but here goes.

Let F_n be the free group of rank n, and Aut(F_n) its automorphism group. How to understand what this group is like? A natural approach is to study its representation theory. But it’s actually not so easy to get a handle on representations of this group. Aut(F_n) acts on F_n^ab = Z^n, so you get one n-dimensional representation; but what else can you find?

The insight of Grunewald and Lubotzky is to consider the action of Aut(F_n) on the homology of interesting finite-index subgroups of F_n. Here’s a simple example: let R be the kernel of a surjection F_n -> Z/2Z. Then R^ab is a free Z-module of rank 2n – 1, and the -1 eigenspace of R^ab has rank n-1. Now F_n may not act on R, but some finite-index subgroup H of F_n does (because there are only finitely many homomorphisms F_n -> Z/2Z, and we can take H to be the stabilizer of the one in question.) So H acquires an action on R^ab; in particular, there is a homomorphism from H to GL_{n-1}(Z). Grunewald and Lubotzky show that this homomorphism has image of finite index in GL_{n-1}(Z). In particular, when n = 3, this shows that a finite-index subgroup of Aut(F_3) surjects onto a finite-index subgroup of GL_2(Z). Thus Aut(F_3) is “large” (it virtually surjects onto a non-abelian free group), and in particular it does not have property T. Whether Aut(F_n) has property T for n>3 is still, as far as I know, unknown.

Grunewald and Lubotzky construct maps from Aut(F_n) to various arithmetic groups via “Prym constructions” like the one above (with Z/2Z replaced by an arbitrary finite group G), and prove under relatively mild conditions that these maps have image of finite index in some specified arithmetic lattice. Of course, it is natural to ask what one can learn from this method about interesting subgroups of Aut(F_n), like the mapping class groups of punctured surfaces. The authors indicate in the introduction that they will return to the question of representations of mapping class groups in a subsequent paper. I very much hope that Lubotzky and others will continue the story that Prof. Grunewald helped to begin.

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