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CATEGORIES:Algebra Seminar
SUMMARY:When Artin groups are sufficiently large... - Sara
h Rees\, University of Newcastle
DTSTART:20140220T160000Z
DTEND:20140220T170000Z
UID:TALK2575AT
URL:/talk/index/2575
DESCRIPTION:An Artin group is a group with a presentation with
generators _x__{1}\,_x__{2}\,...\
,_x_{n}_\, and relations that _x_{ixjxi_... and _xjxixj_... are equal\, where
there are _mi\,j_ terms in the first ex
pression and _mj\,i_ in the second\, fo
r _mi\,j_ ∈ _N_ ∪ {∞}\, _mi\,j_ ≥ 2\, which can be described naturally by a Co
xeter matrix or graph.\n\nThis family of groups co
ntains a wide range of groups\, including braid gr
oups\, free groups\, free abelian groups and much
else\, and its members exhibit a wide range of beh
aviour. Many problems remain open for the family a
s a whole\, including the word problem\, but are s
olved for particular subfamilies. The groups of fi
nite type (mapping onto finite Coxeter groups)\, r
ight-angled type (with each _mi\,j_ ∈ {
2\,∞})\, large and extra-large type (with each _m<
sub>i\,j_ ≥ 3 or 4)\, FC type (every complet
e subgraph of the Coxeter graph corresponds to a f
inite type\nsubgroup) have been particularly studi
ed.\n\nAfter introducing Artin groups and surveyin
g what is known\, I will describe recent work with
Derek Holt and (sometimes) Laura Ciobanu\, dealin
g with a big collection of Artin groups\, containi
ng all the large groups\, which we call 'sufficien
tly large'. For those Artin groups we have element
ary descriptions of the sets of geodesic and shor
tlex geodesic words\, and can reduce any input wor
d to either form. So we can solve the word problem
\, and prove the groups shortlex automatic. And\,
following Appel and Schupp we can solve the conjug
acy problem in extra-large groups in cubic time.\n
\nFor many of the large Artin groups\, including a
ll extra-large groups\, we can deduce the rapid de
cay property and verify the Baum-Connes conjecture
. And although our methods are quite different fro
m those of Godelle and Dehornoy for spherical-type
groups\, we can pool our resources and derive a w
eak form of hyperbolicity for many\, many Artin gr
oups.\n\nI’ll explain some background for the prob
lems we attach\, and outline their solution.
LOCATION:Watson Building\, Lecture Room C
CONTACT:David Craven
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