Narrow Results

We prove that the Torelli group of an oriented surface with any number of boundary components is at least exponentially distorted in the mapping class group by using Broaddus–Farb–Putman’s techniques. Further we show that the distortion of the Torelli group in the level $d$ mapping class group is the same as that in the mapping class group.

In this paper, we consider a certain subgroup ${\mathrm{\text{IA}}}_n^{+}$ of the IA-automorphism group of a free group. We determine the images of the $k$th Johnson homomorphism restricted to ${\mathrm{\text{IA}}}_n^{+}$ for any $k\ge 1$ and $n\ge 2$. By using this result, we give an affirmative answer to the Andreadakis conjecture restricted for ${\mathrm{\text{IA}}}_n^{+}$. Namely, we show that the intersection of the Andreadakis–Johnson filtration and ${\mathrm{\text{IA}}}_n^{+}$ coincides with the lower central series of ${\mathrm{\text{IA}}}_n^{+}$. In a series of this research, we obtain additional results on the integral (co)homology groups of ${\mathrm{\text{IA}}}_n^{+}$. In particular, we determine the first homology group, and study the cup product of first cohomologies of ${\mathrm{\text{IA}}}_n^{+}$. Furthermore, we construct nontrivial second homology classes of ${\mathrm{\text{IA}}}_n^{+}$ by observing its generators and relators, and show that the second cohomology group is not generated by cup products of the first cohomology groups.