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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.

Uday Bhaskar
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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.

Olórin
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  • I know there are papers of Hyman Bass dealing with and defining this (with the definition I gave) but cannot recall the exact papers ATM. Try google maybe. – Olórin Feb 24 '15 at 11:03
  • What does "unique algberaic k-representation" mean? – Angry_Math_Person Mar 23 '22 at 17:22
  • The unique adjective means "being the only one of its kind". Here it means that there's only one $\varphi$ such that $F \circ \varphi = f$. – Olórin Mar 25 '22 at 13:11
  • Actually I was asking what "algberaic representation" means here? Since you wrote it in Italics I'm guessing it's something different from a usual linear representation which is just a homomorphism to $GL(V)$. Is that so? – Angry_Math_Person Mar 26 '22 at 10:21
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    Indeed the algebraic is automatic by the fact that $f$ is what it is. Right now I can't recall, three years after, why I wanted to put an emphasize on algebraic ... hence I remove it. – Olórin Mar 26 '22 at 16:21
  • I'd like to ask another question. Is the inverse limit of affine algberaic groups and algberaic group? As in, does it come with a scheme structure? Your answer seems to suggest that the pro algberaic completion is itself an algberaic group. – Angry_Math_Person Apr 01 '22 at 19:01
  • If the transition maps are group morphism, yes. In general dunno. – Olórin Apr 03 '22 at 12:10
  • An affine algebraic group is the Spec of a $k$-algebra of finite type (endowed with an Hopf algebra structure). Any affine group scheme is an inverse limit of affine algebraic groups. Non algebraic example: $G(k[[t]])$, with $G$ an affine algebraic group over $k$. But it is the inverse limit of the affine algebraic groups $G(k[t] / (t^n))$. – Daniel Juteau May 31 '23 at 12:42