15 research outputs found
Embeddings between partially commutative groups: two counterexamples
In this note we give two examples of partially commutative subgroups of
partially commutative groups. Our examples are counterexamples to the Extension
Graph Conjecture and to the Weakly Chordal Conjecture of Kim and Koberda,
\cite{KK}. On the other hand we extend the class of partially commutative
groups for which it is known that the Extension Graph Conjecture holds, to
include those with commutation graph containing no induced or . In
the process, some new embeddings of surface groups into partially commutative
groups emerge.Comment: 15 pages, 5 figures; to appear in Journal of Algebr
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 satisfying certain graph theoretic conditions
Subgroups of direct products of limit groups over Droms RAAGs
A result of Bridson, Howie, Miller and Short states that if is a subgroup
of type of the direct product of limit groups over
free groups, then is virtually the direct product of limit groups over free
groups. Furthermore, they characterise finitely presented residually free
groups. In this paper these results are generalised to limit groups over Droms
right-angled Artin groups. Droms RAAGs are the right-angled Artin groups with
the property that all of their finitely generated subgroups are again RAAGs. In
addition, we show that the generalised conjugacy problem is solvable for
finitely presented groups that are residually a Droms RAAG and that the
membership problem is decidable for their finitely presented subgroups
Automorphisms of Partially Commutative Groups II: Combinatorial Subgroups
We define several "standard" subgroups of the automorphism group Aut(G) of a
partially commutative (right-angled Artin) group and use these standard
subgroups to describe decompositions of Aut(G). If C is the commutation graph
of G, we show how Aut(G) decomposes in terms of the connected components of C:
obtaining a particularly clear decomposition theorem in the special case where
C has no isolated vertices.
If C has no vertices of a type we call dominated then we give a semi-direct
decompostion of Aut(G) into a subgroup of locally conjugating automorphisms by
the subgroup stabilising a certain lattice of "admissible subsets" of the
vertices of C. We then characterise those graphs for which Aut(G) is a product
(not necessarily semi-direct) of two such subgroups.Comment: 7 figures, 63 pages. Notation and definitions clarified and typos
corrected. 2 new figures added. Appendix containing details of presentation
and proof of a theorem adde
Orthogonal systems in finite grahps
To a finite graph there corresponds a free partially commutative group: with the given graph as commutation graph. In this paper we construct an orthogonality theory for graphs and their corresponding free partially commutative groups. The theory developed here provides tools for the study of the structure of partially commutative groups, their universal theory and automorphism groups. In particular the theory is applied in this paper to the centraliser lattice of such groups
Subgroups of direct products of graphs of groups with free abelian vertex groups
A result of Baumslag and Roseblade states that a finitely presented subgroup
of the direct product of two free groups is virtually a direct product of free
groups. In this paper we generalise this result to the class of cyclic subgroup
separable graphs of groups with free abelian vertex groups and cyclic edge
groups. More precisely, we show that a finitely presented subgroup of the
direct product of two groups in this class is virtually -by-(free abelian),
where is the direct product of two groups in the class. In particular, our
result applies to 2-dimensional coherent right-angled Artin groups and
residually finite tubular groups. Furthermore, we show that the multiple
conjugacy problem and the membership problem are decidable for finitely
presented subgroups of the direct product of two -dimensional coherent
RAAGs