Let A and A be two Artin groups of spherical type, and let A 1 , . . . , A p (resp. A 1 , . . . , A q ) be the irreducible components of A (resp. A ). We show that A and A are commensurable if and only if p = q and, up to permutation of the indices, A i and A i are commensurable for every i. We prove that, if two Artin groups of spherical type are commensurable, then they have the same rank. For a fixed n, we give a complete classification of the irreducible Artin groups of rank n that are commensurable with the group of type A n . Note that it will remain 6 pairs of groups to compare to get the complete classification of Artin groups of spherical type up to commensurability. Proposition 2. Let Γ and Ω be two Coxeter graphs of spherical type. If A[Γ] and A[Ω] are commensurable, then Γ and Ω have the same number of vertices. Proof. Suppose that A[Γ] and A[Ω] are commensurable. Let n be the number of vertices of Γ and let m be the number of vertices of Ω. We know that the cohomological dimension of A[Γ] is n and the cohomological dimension of A[Ω] is m (Paris, 2004, Proposition 3.1). As every finite index subgroup of A[Γ] has the same cohomological dimension as A[Γ] and every finite index subgroup of A[Ω] has the same cohomological dimension as A[Ω], we have n = m.

We show that the geometric realisation of the poset of proper parabolic subgroups of a large-type Artin group has a systolic geometry. We use this geometry to show that the set of parabolic subgroups of a large-type Artin group is stable under arbitrary intersections and forms a lattice for the inclusion. As an application, we show that parabolic subgroups of large-type Artin groups are stable under taking roots and we completely characterise the parabolic subgroups that are conjugacy stable.We also use this geometric perspective to recover and unify results describing the normalizers of parabolic subgroups of large-type Artin groups.

We show that the geometric realisation of the poset of proper parabolic subgroups of a large-type Artin group has a systolic geometry. We use this geometry to show that the set of parabolic subgroups of a large-type Artin group is stable under arbitrary intersections and forms a lattice for the inclusion. As an application, we show that parabolic subgroups of large-type Artin groups are stable under taking roots and we completely characterise the parabolic subgroups that are conjugacy stable. We also use this geometric perspective to recover and unify results describing the normalisers of parabolic subgroups of large-type Artin groups.

In the Cayley graph of the mapping class group of a closed surface, with respect to any generating set, we look at a ball of large radius centered on the identity vertex, and at the proportion among the vertices in this ball representing pseudo-Anosov elements. A well-known conjecture states that this proportion should tend to one as the radius tends to infinity. We prove that it stays bounded away from zero. We also prove similar results for a large class of subgroups of the mapping class group.