Parabolic and quasiparabolic subgroups of free partially commutative groups

Parabolic and Quasiparabolic Subgroups of Free Partially Commutative Groups

Let Γ be a finite graph and G be the corresponding free partially commutative group. In this paper we study subgroups generated by vertices of the graph Γ, which we call canonical parabolic subgroups. A natural extension of the definition leads to canonical quasiparabolic subgroups. It is shown that the centralisers of subsets of G are the conjugates of canonical quasiparabolic centralisers sat...

Automorphisms of Partially Commutative Groups I: Linear Subgroups *

2 Stabilisers of Parabolic Centralisers 6 2.1 Stabiliser Subgroups . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Generators for the Stabiliser of L . . . . . . . . . . . . . . . . 7 2.3 Ordering L and the stabiliser of L-maximal elements . . . . . 9 2.4 Ordering X and closures of elements . . . . . . . . . . . . . . 10 2.5 A matrix representation of St(L) . . . . . . . . . . . . . . . . 1...

Automorphisms of Partially Commutative Groups II: Combinatorial Subgroups

Parabolic subgroups of Garside groups

A Garside monoid is a cancellative monoid with a finite lattice generating set; a Garside group is the group of fractions of a Garside monoid. The family of Garside groups contains the Artin-Tits groups of spherical type. We generalise the well-known notion of a parabolic subgroup of an Artin-Tits group into that of a parabolic subgroup of a Garside group. We also define the more general notion...

Solvability of Equations in Free Partially Commutative Groups Is Decidable

Trace monoids are well-studied objects in computer science where they serve as a basic algebraic tool for analyzing concurrent systems. The question whether the existential theory of trace equations is decidable has been solved positively in 1996. Free partially commutative groups (graph groups) generalize trace monoids in the sense that every element has an inverse. In this paper we show that ...