Topological components of spaces of representations

565

Then we may define a path {xj as follows. Since Г acts transitively on the path components of /~^{уо), there exists уовГ and a path from Xq to Уо^о^- Composing this path with {joxf^}, we obtain a lifted path Xq to x„\. Since /"^Уи,) is path-connected, there exists у^еГ and a path from x^^ to yiX^^. Compose this path with {yixj}«^^^^^^, etc. Continuing in this way we obtain the desired path.

Suppose that Y is path-connected. We claim that Г acts transitively on the path-components of X. Given х,х'бХ, we shall find уеГ and a path Wo^s^i in ^ such that Xo = x and Xi = 7x'. To this end join /(x) and f(x') by a path {yt}o^t^il by the preceding argument there exists a nondecreasing surjective map т: [0, l]-^[0,1] and a path {xjo^sgi such that f{Xs) = y^^s) for O^s^ 1 and Xq = x. Since Г acts transitively on the path components off'^iy^), there exists уеГ such that x^ can be joined by a path inside f~^(yi) to yx'. Composing with the preceding path, we obtain a path from x to x\ proving the claim that Г acts transitively on the path components of X. If Г is trivial, then X is path-connected. П

§2 . Spaces of representations

The purpose of this section is to discuss the general structure of the spaces Hom(7i, G), when тс is a finitely generated group and G is a Lie group. We then specialize to the case when n is the fundamental group of a surface and G is locally isomorphic to PSL(2, R) or P5L(2, C). Invariants are given which distinguish connected components of the spaces Hom(7r, G) in certain cases. The term "connected component" will always refer to connected components in the classical (Hausdorff) topology rather than the Zariski topology, unless otherwise noted.

2 . 1 . Let G denote a Lie group and n a finitely generated group. Suppose that n has a presentation

<^i , . . . , ^ , |Ki ( ^i , . . . , AJ = ...=/>.

Let Hom(7i, G) denote the set of all homomorphisms n-^G. Then the evaluation шар on the generators

ф^ { ф { Л , \ , . . , ф { А^ )

defines a map Hom(7c, G)-^G'", which is injective since {A^, ...,>4„} generates ^. Furthermore its image is the analytic subvariety of G*" consisting of all m- tuples (xi, ...,xj€G'" such that Ri{x^, ...,xj = l for each relation R^. We shall henceforth identify Hom(7i, G) with this analytic variety. If G is a linear algebraic group (with entries in a field X, and defined over a field fe), then Нот (я. G) |s the group of X-points of an algebraic variety defined over k. In particular, if G is a real linear algebraic group, then Hom(7r, G) is a (not necessarily irreduc-