
School of Mathematics and Statistics
Eddy Godelle
University of Sydney
Garside subgroups of Garside groups
Friday 23rd August, 121pm,
Carslaw 373.
Let G^{+} be a monoid and
Delta\in G^{+}. We say that Delta is balanced if
{g\in G^{+}  g\prec \Delta}
= { g\in G^{+}  \Delta \succ g}.
In that case the set of divisors of Delta is denoted
M_\Delta the pair G^{+},\Delta) is a Garside
monoid if G^{+} is Noetherian and cancellative,
Delta is balanced and M_\Delta is a finite
generating set of G^{+} such
(M_\Delta,\prec) and (M_\Delta,\succ) are lattices.
A Garside group is the group of fractions of a Garside monoid. For
instance, braid groups,or more generally Artin groups of spherical
type, are Garside groups. One of the fundamental properties in the
study of Artin groups is the existence of natural subgroups, the
socalled standard parabolic subgroups. Therefore it is natural to ask
what the definition of a standard parabolic subgroup should be in the
context of a Garside group. We will define a family of good candidates
for such subgroups; more generally we will defined what we call
Garside subgroups, which are motivated by several examples and will
investigate some of their properties.
