The genus 2 Torelli group is not finitely generated

The dimension of the Torelli group

“…The classical fact that I(S 1 ) is trivial was proven by Dehn [13] in the 1920s. In 1983, Johnson [29] proved that I(S g ) is finitely generated for g ≥ 3, and in 1986, McCullough-Miller [34] proved that I(S 2 ) is not finitely generated. Mess [37] improved on this in 1992 by proving Theorem D. At the same time, Mess [37] showed that H 3 (I(S 3 ), Z) is not finitely generated (Mess credits the argument to Johnson-Millson).…”

On Genericity of Pseudo-Anosovs in the Torelli Group

“…Recall that the Torelli group I(Σ) is the subgroup of Mod(Σ) consisting of those elements which act trivially on the integer homology of Σ; it is due to Johnson [4] that as long as Σ has at least genus 3, the Torelli group is finitely generated. On the other hand, the Torelli subgroup of the mapping class group is not finitely generated in genus 2 [11] although some of the results we prove still hold in that case.…”

Big Torelli groups: generation and commensuration

“…We record the following observation as a separate lemma, as we will need to make use of it later: Proof. If Σ has finite type, a slick limiting argument due to Feng Luo (see the comment after Proposition 4.6 of [20]) shows that the curve complex has infinite diameter. The obvious adaptation of this method to the case of the Torelli complex also implies that T (Σ) has infinite diameter.…”

“…There is a natural homomorphism MCG(Σ) → Aut(H 1 (Σ, Z)), whose kernel is commonly referred to as the Torelli group I(Σ) < MCG(Σ). While Torelli groups of finite-type surfaces have been the object of intense study (see for example [3,4,12,15,17,20,22,24,26]) not much is known about them in the case of surfaces of infinite type. The present article aims to be a first step in this direction.…”