Search

The hyperelliptic Torelli group is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface and that also commute with some fixed hyperelliptic involution. Putman and the authors proved that this group is generated by Dehn twists about separating curves fixed by the hyperelliptic involution. In this paper, we introduce an algorithmic approach to factoring a wide class of elements of the hyperelliptic Torelli group into such Dehn twists, and apply our methods to several basic types of elements. As one consequence, we answer an old question of Dennis Johnson. [ABSTRACT FROM PUBLISHER]

AbstractIn this paper, we look at the Diophantine equation (0.1)$ n!=\prod_{i=1}^t a_i!\qquad n>a_1\ge a_2\ge \cdots\ge a_t\ge 2. $Under the ABCconjecture, we show that it has only finitely many nontrivial solutions. Unconditionally, we show that the set of nfor which the above equation admits an integer solution a1,.?.?.,atis of asymptotic density zero. [ABSTRACT FROM AUTHOR]

In the paper, we give a lower bound on the genus of an amalgamation of two Heegaard splittings V1, ⋃S1, W1 for M1 and V2, ⋃S2, W2 for M2 along boundary components F1 and F2 under some conditions on the distances of the factor Heegaard splittings. A direct consequence is that under some circumstances, g(M) = g(M1) + g(M2) - g(F) holds, and another is that under some circumstances we have a better lower bound. [ABSTRACT FROM AUTHOR]