TY - JOUR
T1 - On parabolic subgroups of Artin-Tits groups of spherical type
AU - Cumplido, María
AU - Gebhardt, Volker
AU - González-Meneses, Juan
AU - Wiest, Bert
PY - 2019
Y1 - 2019
N2 - We show that, in an Artin–Tits group of spherical type, the intersection of two parabolic subgroups is a parabolic subgroup. Moreover, we show that the set of parabolic subgroups forms a lattice with respect to inclusion. This extends to all Artin–Tits groups of spherical type a result that was previously known for braid groups. To obtain the above results, we show that every element in an Artin–Tits group of spherical type admits a unique minimal parabolic subgroup containing it, which we call its parabolic closure. We also show that the parabolic closure of an element coincides with the parabolic closure of any of its powers or roots. As a consequence, if an element belongs to a parabolic subgroup, all its roots belong to the same parabolic subgroup. We define the simplicial complex of irreducible parabolic subgroups, and we propose it as the analogue, in Artin–Tits groups of spherical type, of the celebrated complex of curves which is an important tool in braid groups, and more generally in mapping class groups. We conjecture that the complex of irreducible parabolic subgroups is δ-hyperbolic.
AB - We show that, in an Artin–Tits group of spherical type, the intersection of two parabolic subgroups is a parabolic subgroup. Moreover, we show that the set of parabolic subgroups forms a lattice with respect to inclusion. This extends to all Artin–Tits groups of spherical type a result that was previously known for braid groups. To obtain the above results, we show that every element in an Artin–Tits group of spherical type admits a unique minimal parabolic subgroup containing it, which we call its parabolic closure. We also show that the parabolic closure of an element coincides with the parabolic closure of any of its powers or roots. As a consequence, if an element belongs to a parabolic subgroup, all its roots belong to the same parabolic subgroup. We define the simplicial complex of irreducible parabolic subgroups, and we propose it as the analogue, in Artin–Tits groups of spherical type, of the celebrated complex of curves which is an important tool in braid groups, and more generally in mapping class groups. We conjecture that the complex of irreducible parabolic subgroups is δ-hyperbolic.
KW - discrete groups
KW - group theory
UR - http://handle.westernsydney.edu.au:8081/1959.7/uws:52420
U2 - 10.1016/j.aim.2019.06.010
DO - 10.1016/j.aim.2019.06.010
M3 - Article
SN - 0001-8708
VL - 352
SP - 572
EP - 610
JO - Advances in Mathematics
JF - Advances in Mathematics
ER -