Proalgebraic completion.

Question

For a finitely generated group, say $\Gamma$, what is the meant by of the proalgebraic completion of $\Gamma$? I came across this while seeing a paper on Representation Growth for Linear Groups by Larsen and Lubotzky.

Answer 1

Let $k$ be a algebraically closed field of characteristic $0$. (One usually takes $k=\mathbf{C}$ but it is not important for the definition.) A (and in fact the) pro-algebraic completion $\widehat{\Gamma}^{\textrm{Alg.}}$ of $\Gamma$ is a pro-algebraic (meaning by that inverse limit of (of a projective system of) algebraic groups) having the following universal property : the algebraic group $\widehat{\Gamma}^{\textrm{Alg.}}$ comes equipped with a morphism of groups $\varphi : \Gamma \to \widehat{\Gamma}^{\textrm{Alg.}}$ such that for any finite-dimensional $k$-representation $f : \Gamma \to \textrm{GL}(V)$ of $\Gamma$ there existe a unique $k$-representation $F : \widehat{\Gamma}^{\textrm{Alg.}} \to \textrm{GL}(V)$ of $A(\Gamma)$ such that $F \circ \varphi = f$.

As far as I remember, $\widehat{\Gamma}^{\textrm{Alg.}}$ is also called the Hochschild-Mostow group of $\Gamma$. You could have a look at Hochschild's and Mostow's Representations and representative functions of Lie groups* in Annals of Mathematics, second series, number 66.