Spherical quadratic equations in free metabelian groups

Abstract

Abstract. We prove that the Diophantine problem for spherical quadratic equations in free metabelian groups is solvable and, moreover, NP-complete.

Quadratic equations in the Grigorchuk group

“…This line of research for free groups was continued: solution sets were studied in [9], NPcompleteness was proved in [7,14]. Various classes of infinite groups were analyzed: hyperbolic groups were studied in [10,15], the first Grigorchuk group was studied in [17] and [1], free metabelian groups were studied in [18,19], metabelian Baumslag-Solitar groups were studied in [21]. The Diophantine problem for quadratic equations over the lamplighter group L 2 was recently shown to be decidable by Kharlampovich, Lopez, and Miasnikov (see [13]).…”

The winding invariant

“…, z n ) → H which maps w to 1 ∈ H. It is worth mentioning that the general solvability problem for equations in the free metabelian group of rank 2 is algorithmically unsolvable [138]. However, for some quadratic equations the problem is solvable [107,108] and other problems over metabelian groups are known to be algorithmically solvable (see [19], [22] and [94, Section 9.5], for example).…”

Magnus embedding and algorithmic properties of groups 𝐹/𝑁^{(𝑑)}

“…In [18] those complexity bounds were further improved and randomized algorithms were developed. Another generalization was done by Lysenok and Ushakov in [6]. It was shown that the Diophantine problem for spherical quadratic equations, i.e., equations of the form: z −1 1 c 1 z 1 .…”

“…We also would like to mention several results related to practical computations in free solvable groups in which the Magnus embedding plays a crucial role. S. Vassileva showed in [19] that the power problem in free solvable groups can be solved in O(rd(|u| + |v|) 6 ) time and used that result to show that the Matthews-Remeslennikov-Sokolov approach can be transformed into a polynomial time O(rd(|u|+ |v|) 8 ) algorithm for the conjugacy problem. In [18] those complexity bounds were further improved and randomized algorithms were developed.…”