Presentations for rigid solvable groups

Abstract

Abstract. A group G is said to be m-rigid if it has a normal seriesin which each factor G i =G i C1 is abelian and torsion-free as a ZOEG=G i -module. Denote by † m the class of all m-rigid groups and by † m .R/ the set of groups in † m generated by x 1 ; : : : ; x n that satisfy a given set of relations R. We say that a group in † m .R/ is maximal if it has no proper covering in † m .R/. It is proved that, for every R, the set † m .R/ contains only finitely many maximal groups. The set of relations R is said to be complete if † m .R/ contains a unique maximal group. It is shown that every finitely generated group in † m is completely finitely presented. We give a definition of a canonical presentation for a rigid group with the generators x 1 ; : : : ; x n . If such a presentation is given, the group at least has decidable word problem. Given a finite set of relations R D R.x 1 ; : : : ; x n /, we effectively construct a finite set of canonical presentations in the generators x 1 ; : : : ; x n for groups in † m .R/ among which all the maximal groups in † m .R/ are contained.

Algebraic Sets in a Finitely Generated 2-Step Solvable Rigid Pro-p-Group

“…Abstract rigid solvable groups and algebraic geometry over them were defined and studied in [1][2][3][4][5][6][7]. In [8], the notion of a rigid metabelian pro-p-group was defined and some properties of such groups were explored.…”

Rigid Metabelian Pro-p-Groups

“…In [1][2][3][4][5][6][7], rigid solvable groups were defined and explored, and many aspects of algebraic geometry over such groups were studied. Important examples of rigid groups are free solvable groups.…”

Automorphisms of Divisible Rigid Groups

“…A rigid group is one that is m-rigid for some m. Rigid groups were introduced and explored in [1][2][3][4][5][6][7][8].…”