Vol 85, No 6 (2021)

The Torelli group of a closed oriented surface $S_g$ of genus $g$ is the subgroup$\mathcal{I}_g$ of the mapping class group $\operatorname{Mod}(S_g)$ consisting ofall mapping classes that act trivially on the homology of $S_g$. One of the most intriguingopen problems concerning Torelli groups is the question of whether the group $\mathcal{I}_3$is finitely presented. A possible approach to this problem relies on the study of the secondhomology group of $\mathcal{I}_3$ using the spectral sequence $E^r_{p,q}$ for the actionof $\mathcal{I}_3$ on the complex of cycles. In this paper we obtain evidence forthe conjecture that $H_2(\mathcal{I}_3;\mathbb{Z})$ is not finitely generated and hence$\mathcal{I}_3$ is not finitely presented. Namely, we prove that the term $E^3_{0,2}$ ofthe spectral sequence is not finitely generated, that is, the group $E^1_{0,2}$ remainsinfinitely generated after taking quotients by the images of the differentials $d^1$ and $d^2$.Proving that it remains infinitely generated after taking the quotient by the imageof $d^3$ would complete the proof that $\mathcal{I}_3$ is not finitely presented.

This paper is the first in a series of three devoted to constructing a finitely presented infinite nilsemigroup satisfying the identity $x^9=0$. This solves a problem of Lev Shevrin and Mark Sapir.In this first part we obtain a sequence of complexes formed of squares ($4$-cycles) having the following geometric properties.1) Complexes are uniformly elliptic. A space is said to be uniformly elliptic if there is a constant $\lambda>0$ such that in the set of shortest paths of length $D$ connecting points $A$ and $B$ there are two paths such that the distance between them is at most $\lambda D$. In this case, the distance between paths with the same beginning and end is defined as the maximal distance between the corresponding points. 2) Complexes are nested.A complex of level $n+1$ is obtained from a complex of level $n$ by adding several vertices and edges according to certain rules.3) Paths admit local transformations. Assume that we can transform paths by replacing a path along two sides of a minimal square by the path along the other two sides. Two shortest paths with the same ends can be transformed into each other locally if these ends are vertices of a square in the embedded complex.The geometric properties of the sequence of complexes will be further used to define finitely presented semigroups.

In this paper we study the Diophantine problem in the classical matrix groups $\mathrm{GL}_n(R)$, $\mathrm{SL}_n(R)$, $\mathrm{T}_n(R)$ and $\mathrm{UT}_n(R)$, $n \geq 3$, over an associative ring $R$ with identity. We show that if $G_n(R)$ is one of these groups, then the Diophantine problem in $G_n(R)$ is polynomial-time equivalent (more precisely, Karp equivalent) to the Diophantine problem in $R$. When $G_n(R)=\mathrm{SL}_n(R)$ we assume that $R$ is commutative. Similar results hold for $\mathrm{PGL}_n(R)$ and $\mathrm{PSL}_n(R)$ provided $R$ has no zero divisors (for $\mathrm{PGL}_n(R)$ the ring $R$ is not assumed to be commutative).