Universal Upper Bound for the Growth of Artin Monoids

Abstract

ABSTRACT. In this paper we study the growth rates of Artin monoids and we show that 4 is a universal upper bound. We also show that the generating functions of the associated right-angled Artin monoids are given by families of Chebyshev polynomials. Applications to Artin groups and positive braids are given. .

Recurrence Relations and Hilbert Series of the Monoid Associated with Star Topology

“…In 1993, Parry [9] gave the growth series of Coxeter groups. In [10], we proved that the upper bound of the growth of spherical Artin monoids is 4. But, in the affine case, this result is not true.…”

Some Recurrence Relations and Hilbert Series of Right-Angled Affine Artin Monoid M(D~n∞)

“…These well-known affine Coxeter groups arẽ,̃,̃,̃,̃6,̃7,̃8,̃2, and̃1 (for details, see [5]). In [3] authors proved that the universal upper bound for all the spherical Artin monoids is less than 4.…”

“…In [2] Iqbal and Yousaf computed the Hilbert series of the braid monoid 4 in band generators. In [3] Berceanu and Iqbal proved that the growth rate of all the spherical Artin monoids is less than 4. In [4] Iqbal et al studied the braid monoid (̃∞) of the affine typẽof the Coxeter systems.…”