# Homomorphisms of the free group $F_n$ to $GL_k(\mathbb{R})$ [closed]

Is there something known about the set of all homomorphisms from the free group on $n$ generators $F_n$ to the real general linear group $GL_k(\mathbb{R})$ modulo conjugation, i.e.

$$Hom(F_n, GL_k(\mathbb{R}))/GL_k(\mathbb{R}) ,$$

or, differently formulated, of the set $\{ (A_1, \ldots, A_n)| A_i \in GL_k(\mathbb{R}) \} / GL_k(\mathbb{R})$?

• What kind of something do you have in mind? As you have observed, it is the set of simultaneous conjugacy classes of $n$-tuples. – HJRW Jan 24 '11 at 17:55
• You might find the complex case interesting. This has been studied by Sean Lawton and Carlos Florentino (Math. Ann. 345, 2009, 453-489). They show that the categorical quotient Hom$(F_n, GL_k C)//GL_k C$ is homotopy equivalent to Hom$(F_n, U(n))/U(n)$. – Dan Ramras Jan 24 '11 at 19:33
• Whoops, $U(n)$ should be $U(k)$ in the above comment. – Dan Ramras Jan 24 '11 at 19:34
• This question appears to be off-topic because it is no longer relevant, the question was answered to the extent possible. – Misha Oct 19 '13 at 12:23

## 3 Answers

As Henry Wilton asked, it would be nice to know what something you would like to know. If you restrict to $GL_+(k,\mathbb R)$, meaning elements with positive determinant, then you can split the matrices as positive multiple of the identity times a matrix of determinant 1, so a good place to start is to look at homomorphisms to $SL(k,\mathbb R)$.

In general, the space of homomorphisms up to conjugacy is not Hausdorff. For example, when $k=2, n=1$, the matrices $((1,a),(0,1))$ are all conjugate for $a \ne 0$, but they limit to the identity matrix. But most of these quotients are Hausdorff, and in any case one can take the Hausdorffication. For $k=2$, the trace of an element of $SL(2,\mathbb R)$ is a complete invariant of the Hausdorfication; in general, you need the coefficients of the characteristic polynomial, or equvialently, the sequence of traces of the first $n-1$ powers to parametrize the Hausddorffication of the conjugacy class of an element of $SL(n,\mathbb R)$.

For $k=2, n=2$, if the generators of the free group map to elements $A$ and $B$, then the traces of $A, B, AB$ are enough to determine the Hausdorffication of the set of these homomorphisms. The trace of any other element in the group they generate is a polynomial in these three traces, which can be chosen arbitrarily and independently. These can be found using the trace relation $$T(AB) + T(A B^{-1}) = T(A) T(B)$$ (This can be derived from the fact that $A$ satisfies its characteristic equation, $A^2- T(A) A + I = 0$, multiplying on the right by $A^{-1} B$ and taking the trace.)

For $k=2, n=3$, the traces of $A, B, C, AB, BC, CA, ABC$ are enough to determine the conjugacy class, but the trace of $ABC$ satisfies a quadratic equation with coefficients that are polynomials in the other traces. (The second root is the trace of $ACB$).

For general $k$ and $n$, the ring generated by all traces of all elements of the group (as a function on the space of homomorphisms from the free group to $SL(n,\mathbb R)$, or to $GL(n, \mathbb R)$ is generated by the traces of a finite subset of the elements; it's something people have studied, it gets more and more complicated, but I don't know the details.

There is a vast literature on the subject, but as @Henry asks: what exactly do you care about? In the particular case of $k=2$ you might want to look at the papers of Bill Goldman (one example is: MR2497777 (2010j:30093) Goldman, William M.(1-MD) Trace coordinates on Fricke spaces of some simple hyperbolic surfaces. Handbook of Teichmüller theory. Vol. II, 611–684, IRMA Lect. Math. Theor. Phys., 13, Eur. Math. Soc., Zürich, 2009. ), but this subject is of positive density among his publications.

Vinberg has done work for higher $k$ from a more invariant-theoretic standpoint. You should look at his papers on invariant theory (I remember a talk of his around 20 years ago where he was talking about complete sets of invariants of free subgroups of $GL_k,$ as an analogue of the fact that for $k=n=2$ the representation is determined up to conjugacy by $tr(A), tr(B), tr(AB),$ where $A, B$ are the images of the free generators. I can't figure out which of his papers this is, though).

To help with searching: the set you ask about is usually called the "representation variety" or the "character variety." The study of representation varieties of surface groups (of which free groups are examples) is, as others have said, very active. These lecture notes of Richard Wentworth seem a good place to start.

One topic of special interest is the action of Aut(F_n) on your set by composition on the left -- the key word here is "Product Replacement Algorithm," as in the work of Igor Pak. Perhaps even more keenly studied is the action of the mapping class group of the surface, which is contained in Aut(F_n) (which subgroup it is depends on which open surface of Euler characteristic 2-n you have in mind.)