Daniel K. Biss

K_g is not finitely generated

We prove that for any genus g>1, the subgroup K_g of the mapping class group of a closed genus g surface generated by Dehn twists about separating curves is not finitely generated.

Large Annihilators in Cayley–Dickson Algebras

Cayley–Dickson algebras are nonassociative ℝ-algebras that generalize the well-known algebras ℝ, ℂ, ℍ, and 𝕆. We study zero-divisors in these algebras. In particular, we show that the annihilator of any element of the 2 n -dimensional Cayley–Dickson algebra has dimension at most 2 n − 4n + 4. Moreover, every multiple of 4 between 0 and this upper bound occurs as the dimension of some annihilator. Although a complete description of zero-divisors seems to be out of reach, we can describe precisely the elements whose annihilators have dimension 2 n − 4n + 4.

Eigentheory of Cayley-Dickson algebras

Abstract We show how eigentheory clarifies many algebraic properties of Cayley-Dickson algebras. These notes are intended as background material for those who are studying this eigentheory more closely.