623 research outputs found
Automorphism groups of some affine and finite type Artin groups
We observe that, for each positive integer n > 2, each of the Artin groups of
finite type A_n, B_n=C_n, and affine type \tilde A_{n-1} and \tilde C_{n-1} is
a central extension of a finite index subgroup of the mapping class group of
the (n+2)-punctured sphere. (The centre is trivial in the affine case and
infinite cyclic in the finite type cases). Using results of Ivanov and Korkmaz
on abstract commensurators of surface mapping class groups we are able to
determine the automorphism groups of each member of these four infinite
families of Artin groups
The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group
It was conjectured by Tits that the only relations amongst the squares of the
standard generators of an Artin group are the obvious ones, namely that a^2 and
b^2 commute if ab=ba appears as one of the Artin relations. In this paper we
prove Tits' conjecture for all Artin groups. More generally, we show that,
given a number m(s)>1 for each Artin generator s, the only relations amongst
the powers s^m(s) of the generators are that a^m(a) and b^m(b) commute if ab=ba
appears amongst the Artin relations.Comment: 18 pages, 11 figures (.eps files generated by pstricks.tex).
Prepublication du Laboratoire de Topologie UMR 5584 du CNRS (Univ. de
Bourgogne
Boundary quotients and ideals of Toeplitz C*-algebras of Artin groups
We study the quotients of the Toeplitz C*-algebra of a quasi-lattice ordered
group (G,P), which we view as crossed products by a partial actions of G on
closed invariant subsets of a totally disconnected compact Hausdorff space, the
Nica spectrum of (G,P). Our original motivation and our main examples are drawn
from right-angled Artin groups, but many of our results are valid for more
general quasi-lattice ordered groups. We show that the Nica spectrum has a
unique minimal closed invariant subset, which we call the boundary spectrum,
and we define the boundary quotient to be the crossed product of the
corresponding restricted partial action. The main technical tools used are the
results of Exel, Laca, and Quigg on simplicity and ideal structure of partial
crossed products, which depend on amenability and topological freeness of the
partial action and its restriction to closed invariant subsets. When there
exists a generalised length function, or controlled map, defined on G and
taking values in an amenable group, we prove that the partial action is
amenable on arbitrary closed invariant subsets. Our main results are obtained
for right-angled Artin groups with trivial centre, that is, those with no
cyclic direct factor; they include a presentation of the boundary quotient in
terms of generators and relations that generalises Cuntz's presentation of O_n,
a proof that the boundary quotient is purely infinite and simple, and a
parametrisation of the ideals of the Toeplitz C*-algebra in terms of subsets of
the standard generators of the Artin group.Comment: 26 page
Embeddings of graph braid and surface groups in right-angled Artin groups and braid groups
We prove by explicit construction that graph braid groups and most surface
groups can be embedded in a natural way in right-angled Artin groups, and we
point out some consequences of these embedding results. We also show that every
right-angled Artin group can be embedded in a pure surface braid group. On the
other hand, by generalising to right-angled Artin groups a result of Lyndon for
free groups, we show that the Euler characteristic -1 surface group (given by
the relation x^2y^2=z^2) never embeds in a right-angled Artin group.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-22.abs.htm
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