nLab Makanin-Razborov diagram

Idea

Given a system of finitely many word equations

$$F_\alpha(a_1,\ldots,a_n,x_1,\ldots,x_m) = 1 \,, \;\;\; \text{where}\; \alpha \in \{1,\ldots, K\}$$

with coefficients $a_1,\ldots, a_n$ in a group $G$, a solution in $G$ for unknowns $x_1,\ldots,x_m$ supplies the evaluation “functional” in $Hom(G_F,G)$, where $G_F$ is the direct product of $n+m$ copies of $G$ modulo the normal subgroup determined by $F_\alpha$ for fixed constants $a_i$.

Makanin 1982 found an algorithm which for such a finite system of equations over a free group on $s$ letters produces all solutions when at least one exists. In his thesis, Razborov 1984 developed the theory further.

Later, similar schemes, now called Makanin-Razborov diagrams, for finding $Hom(H,G)$ in the larger generality where $H$ is some finitely generated group, were discussed by various authors, including Sela, Reinfeldt, Weidmann, Bestvina, and Feighn.

Literature

The original articles:

• Г. С. Маканин, Уравнения в свободной группе, Изв. АН СССР. Сер. матем., 46:6 (1982), 1199–1273 pdf; engl. transl. G. S. Makanin, Equations in a free group,, Math. USSR-Izv., 21:3 (1983), 483–546 pdf

• А. А. Разборов, О системах уравнений в свободной группе, Изв. АН СССР. Сер. матем., 48:4 (1984), 779–832; engl. transl. A. A. Razborov, On systems of equations in a free group, Math. USSR-Izv., 25:1 (1985), 115–162

Review:

• Richard Weidmann, Cornelius Reinfeldt: Makanin–Razborov diagrams for hyperbolic groups, Annales mathématiques Blaise Pascal 26 2 (2019) 119–208 [doi:10.5802/ambp.387]

• Montserrat Casals-Ruiz, Ilya Kazachkov: Makanin-Razborov diagrams, talk notes (2016) [pdf]