arXiv subjects

Alden Walker

Publications and source records attributed to Alden Walker.

21 records · Page 2Linked to original sources

arXiv2011

Ziggurats and rotation numbers

We establish the existence of new rigidity and rationality phenomena in the theory of nonabelian group actions on the circle, and introduce tools to translate questions about the existence of actions with prescribed dynamics into finite combinatorics. A special case of our theory gives a very short new proof of Naimi's theorem (i.e. the conjecture of Jankins-Neumann) which was the last step in the classification of taut foliations of Seifert fibered spaces.

math.DS↗

arXiv2011

Isometric endomorphisms of free groups

An arbitrary homomorphism between groups is nonincreasing for stable commutator length, and there are infinitely many (injective) homomorphisms between free groups which strictly decrease the stable commutator length of some elements. However, we show in this paper that a random homomorphism between free groups is almost surely an isometry for stable commutator length for every element; in particular, the unit ball in the scl norm of a free group admits an enormous number of exotic isometries. Using similar methods, we show that a random fatgraph in a free group is extremal (i.e. is an absolute minimizer for relative Gromov norm) for its boundary; this implies, for instance, that a random element of a free group with commutator length at most n has commutator length exactly n and stable commutator length exactly n-1/2. Our methods also let us construct explicit (and computable) quasimorphisms which certify these facts.

math.GR↗

arXiv2009

Product Structures for Legendrian Contact Homology

Legendrian contact homology (LCH) and its associated differential graded algebra are powerful non-classical invariants of Legendrian knots. Linearization makes the LCH computationally tractable at the expense of discarding nonlinear (and noncommutative) information. To recover some of the nonlinear information while preserving computability, we introduce invariant cup and Massey products - and, more generally, an A_\infty structure - on the linearized LCH. We apply the products and A_\infty structure in three ways: to find infinite families of Legendrian knots that are not isotopic to their Legendrian mirrors, to reinterpret the duality theorem of the fourth author in terms of the cup product, and to recover higher-order linearizations of the LCH.

math.SG↗