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 ?

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

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