What can be the applications of a theory of schemes à la Grothendieck to the category of groups?

Question

Baumslag, Miasnikov, and Remeslennikov have developed in [1] a theory similar to the classical theory of algebraic geometry in the category of groups. Let $G$ be a group, a $G$-group is defined by a couple $(H,f_H)$ where $f_H:G\rightarrow H$ is an (injective) morphism. For every element $x\in H$, we denote by $G(x)$, the subgroup of $H$ generated by $\{f_H(g)xf_H(g)^{-1}, g\in G\}$. An element $x\neq 1$ of $H$ is a divisor of zero if there exists an element $y\neq 1$ such that $[G(x),G(y)]=1$. A $G$-group $H$ is a domain if there does not exist a divisor of zero in $H$. An ideal $I$ of $(H,f_H)$ is just a normal subgroup of $H$ distinct of $H$. If $I$ of $H$, $(H/I,f_{H/I})$ is a $G$-group where $f_{H/I}$ is the composition of $f_H$ and the quotient morphism $H\rightarrow H/I$. The ideal $P$ is a prime ideal if and only if $(H/I,f_{H/I})$ is a domain. This is equivalent to saying that for every $x,y\in H$, $[G(x),G(y)]\subset P$ implies that $x\in P$ or $y\in P$.

Let $G[X_1,...,X_n]$ be the free product of $G$ and the free group generated by $n$ elements. Every element $f$ of $G[X_1,...,X_n]$ defines a map $f_P:H^n\rightarrow H$ such that $f_P(h_1,..,h_n)$ is obtained by replacing $X_i$ by $h_i$ in the expression of $f$. Such a function is called a polynomial function. For every subset $S$ of $G[X_1,...,X_n]$, $V_H(S)=\{x\in H^n:\forall f\in S, f_P(x)=1\}$. The Zariski topology of $H^n$ is the topology whose set of subsets is generated by $\{S\subset G[X_1,...,X_n],V_H(S)\}$.

I have started to adapt the point of view of Grothendieck to this theory. Here I consider the comma category $C(G)$ whose objects are morphisms $f_H:G\rightarrow H$.Let $Spec_G(H)$ be the set of prime ideals of $H$. For every normal subgroup $I$ of $H$, $V_H(I)$ is the set of prime which contains $I$. I have noticed that $V_H([I,J])=V_H(I)\bigcup V_H(J)$ and for a family of normal subgroup $(I_a)_{a\in A}$, let $E_A$ be the group generated by the family $(I_a)_{a\in A}$, $V(E_a)=\cap_{a\in A}V(I_a)$. This endows $Spec_G(H)$ with a topology. Let $U$ be an open subset of $Spec_G(H)$, we denote by $O_H(U)$ the set of functions defined on $U$ such that for every $f\in O_H(U)$, for every $P\in U$, there exists an open subset $V$ of $U$ which contains $P$, an element $h_V\in H$ such that for every $Q\in V$, $f(Q)=l_Q(h_V)$ where $l_Q:H\rightarrow H/Q$ is the quotient map. The correspondence $U\rightarrow O_H(U)$ defines a sheaf on $Spec_G(H)$.

A $G$-scheme $(X,O_X)$ is a topological space $X$ endowed with a sheaf $O_X$ such that there exists an open cover $(U_i)_{i\in I}$ of $X$, for every $i\in I$ a $G$-group $H_i$ such that there exists an isomorphism of $G$-spaces between $(U_i,{O_X}_{U_i})$ and $(Spec_G(H_i),O_{H_i})$. See the second reference for more details.

We can also define another notion of prime by saying that an ideal $I$ is a prime if and only if for every $x,y\in H$ such that $G(x)\cap G(y)\subset I$, $x\in I$ or $y\in I$ and develop a similar theory.

Question.

The theory of schemes has very important applications in algebraic geometry, what are the problems in group theory which can be solved by using this framework ?

Let $G$ be a group, we can study the spectrum $Spec_G(G)$. Suppose that $G$ is the free group generated by $n$ elements. A closed subset of $Spec_G(G)$ is defined by a normal subgroup $I$ of $G$ which is the presentation of a finitely generated group. What can be the interpretation of the topology of $Spec_G(G)$ in this case ? Is anyone familiar with this construction ?

1. Baumslag, G, Miasnikov, A. Remeslennikov, V.N. Algebraic geometry over groups I. Algebraic sets and ideal theory. J. Algebra. 1999, 219, 1679.

2. Tsemo Aristide. Some properties of G-schemes. http://xxx.lanl.gov/pdf/1708.00359v1

Answer 1

The topology you discuss is closely related to the Gromov--Grigorchuk topology on the space of marked groups. You might be interested in this nice paper of Champetier--Guirardel.

In recent years, most of the work on this topic has been aimed at the case when $G$ is free. (Indeed, I think this is what Baumslag et al. had in mind.) The culmination of this work is Zlil Sela's solution to Tarski's problems on the elementary theory of the free group, and the parallel project of Kharlampovich--Myasnikov.

As I think you'll see in the Champetier--Guirardel paper, for specific groups $G$, the "soft" Grothendieck-ian point of view encompassed by the topology on $\mathrm{Spec}(G)$ doesn't get you very far -- I think the point is that non-commutative groups are sufficiently "hard" that even basic properties don't hold in general, and so are inaccessible to "soft" techniques.

To give you some idea of the techniques involved, Sela's work hinges entirely on the fact that a free group $F$ acts freely on a tree. He needs to analyse infinite sequences of homomorphisms $f_i:H\to F$ (I hope my $H$ is the same as yours, but am not sure) and he always does so by passing to a limiting action of $H$ on a real tree, and analysing the dynamics of this action. All very un-Grothendieck-ian! Kharlampovich--Myasnikov's techniques are superficially more combinatorial, but morally amount to much the same thing.