Subgroups of Teichmüller Modular Groups

Free subgroups of surface mapping class groups

“…This recovers the theorem of Ivanov [12] and McCarthy [16]. In Section 5 we shall give a quantitative version of this result.…”

Injections of Artin groups

“…Superinjective maps are easily seen to be injective (given two vertices, there is a vertex connected to one but not the other), so it follows that superinjective maps are automorphisms. We can then apply the theorem of Ivanov, Korkmaz, and Luo that if S is any surface of negative Euler characteristic other than S 1,2 , then each automorphism of C(S) is induced by Mod(S) [22], [26], [28]. The result is that, for these surfaces, superinjective maps of C(S) are induced by Mod(S); this is the general case of Theorem 2.…”

“…This was first proven by DyerGrossman [14]. Ivanov was the first to compute Aut(A(A n )) from the perspective of mapping class groups [22].…”

“…In brief, Ivanov first proved the automorphism theorem for surfaces of genus at least 2 [22], then Korkmaz proved it for genus 0 and 1 [26]. Around the same time, Luo gave a completely different proof for all surfaces, and also settled the case of the twice punctured torus [28].…”

Surface subgroups of mapping class groups

“…In particular, both the finite and infinite index subgroup structure of Γ g has many parallels in the theory of lattices. For example, the question of whether Property T holds for Γ g ≥ 3 (it fails for g = 2 by [29]), whether Γ g has a version of the Congruence Subgroup Property, or towards the other extreme, whether there are finite index subgroups of Γ g that surject onto Z (see for example [11], [16], [18] and [29] for more on these directions). The discussion and questions raised in this paper are motivated by analogies between the subgroup structure of Γ g and non-cocompact but finite co-volume Kleinian groups.…”