We characterize the fractional Dehn twist coefficient (FDTC) on the n-stranded braid group as the unique homogeneous quasimorphism to $\mathbb {R}$ of defect at most 1 that equals 1 on the positive full twist and vanishes on the $(n-1)$-stranded braid subgroup. In a different direction, we establish that the slice-Bennequin inequality holds with the FDTC in place of the writhe. In other words, we establish an affine linear lower bound for the smooth slice genus of the closure of a braid in terms of the braid’s FDTC. We also discuss connections between these two seemingly unrelated results. In the appendix, we provide a unifying framework for the slice-Bennequin inequality and its counterpart for the FDTC.

We introduce “braided” versions of self-similar groups and Röver–Nekrashevych groups, and study their finiteness properties. This generalizes work of Aroca and Cumplido, and the first author and Wu, who considered the case when the self-similar groups are what we call “self-identical.” In particular, we use a braided version of the Grigorchuk group to construct a new group called the “braided Röver group,” which we prove is of type $\operatorname {\mathrm {F}}_\infty $. Our techniques involve using so-called d-ary cloning systems to construct the groups, and analyzing certain complexes of embedded disks in a surface to understand their finiteness properties.

We develop a method based on the Burau matrix to detect conditions on the linking numbers of braid strands. Our main application is to iterated exchanged braids. Unless the braid permutation fixes both braid edge strands, we establish under some fairly generic conditions on the linking numbers a ‘subsymmetry’ property; in particular at most two such braids can be mutually conjugate. As an addition, we prove that the Burau kernel is contained in the commutator subgroup of the pure braid group. We discuss also some properties of the Burau image.

The Burau representation of the braid group can be used to recover the Alexander polynomial of the closure of a braid. We define twisted Burau maps and use them to compute twisted Alexander polynomials.

Let C be a proper smooth geometrically connected hyperbolic curve over a field of characteristic 0 and ℓ a prime number. We prove the injectivity of the homomorphism from the pro-ℓ mapping class group attached to the two dimensional configuration space of C to the one attached to C, induced by the natural projection. We also prove a certain graded Lie algebra version of this injectivity. Consequently, we show that the kernel of the outer Galois representation on the pro-ℓ pure braid group on C with n strings does not depend on n, even if n = 1. This extends a previous result by Ihara–Kaneko. By applying these results to the universal family over the moduli space of curves, we solve completely Oda's problem on the independency of certain towers of (infinite) algebraic number fields, which has been studied by Ihara, Matsumoto, Nakamura, Ueno and the author. Sequentially we obtain certain information of the image of this Galois representation and get obstructions to the surjectivity of the Johnson–Morita homomorphism at each sufficiently large even degree (as Oda predicts), for the first time for a proper curve.

We construct a Banach space X whose algebra of bounded linear operators has K0-group the additive group of rational numbers. We achieve this using a Gowers-Maurey space.

2000 Mathematical Subject Classification: 20F36, 55P35, 55Q05, 55Q40, 55U10

By studying the braid group action on Milnor's construction of the 1-sphere, we show that the general higher homotopy group of the 3-sphere is the fixed set of the pure braid group action on certain combinatorially described groups. This establishes a relation between the braid groups and the homotopy groups of the sphere.

2000Mathematical Subject Classification: 20F36, 55P35, 55Q05, 55Q40, 55U10.