17:00
I’ll report on my recent work (with co-authors Holt and Ciobanu) on Artin
groups of large type, that is groups with presentations of the form
G = hx1, . . . , xn | xixjxi · · · = xjxixj · · · , 8i 3. (In fact, our results still hold when some, but not all
possible, relations with mij = 2 are allowed.)
Recently, Holt and I characterised the geodesic words in these groups, and
described an effective method to reduce any word to geodesic form. That
proves the groups shortlex automatic and gives an effective (at worst quadratic)
solution to the word problem. Using this characterisation of geodesics, Holt,
Ciobanu and I can derive the rapid decay property for most large type
groups, and hence deduce for most of these that the Baum-Connes conjec-
ture holds; this has various consequence, in particular that the Kadison-
Kaplansky conjecture holds for these groups, i.e. that the group ring CG
contains no non-trivial idempotents.
1