Alden Walker

INTEGER HULLS OF LINEAR POLYHEDRA AND SCL IN FAMILIES

The integer hull of a polyhedron is the convex hull of the integer points contained in it. We show that the vertices of the integer hulls of a rational family of polyhedra of size O(n) have quasipolynomial coordinates. As a corollary, we show that the stable commutator length of elements in a surgery family is a ratio of quasipolynomials, and that unit balls in the scl norm quasi-converge in finite dimensional surgery families.

Surface subgroups from linear programming

We show that a random endomorphism of a free group gives rise to an HNN extension which contains a closed surface subgroup, with probability one; a special case is the HNN extension associated to the endomorphism of a rank 2 free group (considered by Sapir) sending a to ab and b to ba. We further show that a group obtained by doubling a free group along any collection of subgroups (of arbitrary rank) has the property that every rational 2-dimensional homology class is virtually represented by surface subgroups, and the unit ball in the Gromov norm is a finite sided rational polyhedron. These results are obtained by a mixture of combinatorial, geometric and linear programming techniques. We obtain other related results of a more technical nature concerning stable commutator length in free groups, especially concerning the structure of the space of so-called counting quasimorphisms

Random groups contain surface subgroups

1. IntroductionGromov famously asked the following:Surface Subgroup Question. Let G be a one-ended hyperbolic group. Does Gcontain a subgroup isomorphic to the fundamental group of a closed surface withχ < 0?Beyond its intrinsic appeal, and its obvious connections to the Virtual HakenConjecture in 3-manifold topology (now a theorem of Agol [1]), one reason Gromovwas interested in this question was the hope that such surface subgroups could beused as essential structural components of hyperbolic groups [9]. Our interest inthis question is stimulated by a belief that surface groups (not necessarily closed)can act as a sort of “bridge” between hyperbolic geometry and symplectic geometry(through their connection to causal structures, quasimorphisms, stable commutatorlength, etc.).Despite receiving considerable attention the Surface Subgroup Question is wideopen in general, although in the specific case of hyperbolic 3-manifold groups it waspositively resolved by Kahn–Markovic [10]. The main results of our paper may besummarized by saying that we show that Gromov’s question has a positive answerfor most (hyperbolic) groups. In fact, the “executive summary” says that(1) most groups contain (many) surface subgroups;(2) these surface subgroups are quasiconvex — i.e. their intrinsic and extrinsicgeometry is uniformly comparable on large scales; and(3) these surface subgroups can be constructed, and their properties certifiedquickly and easily.Here “most groups” is a proxy for random groups in Gromov’s few relators ordensity models, to be defined presently.In [8], § 9 (also see [14]), Gromov introduced the notion of a random group. Infact, he introduced two such models: the few relators model and the density model.In either model one first fixes a free group F

Satisfiability-based Set Membership Filters

Introduced here is a novel application of Satisfiability (SAT) to the set membership problem with specific focus on efficiently testing whether large sets contain a given element. Such tests can be greatly enhanced via the use of filters, probabilistic algorithms that can quickly decide whether or not a given element is in a given set. This article proposes SAT filters (i.e., filters based on SAT) and their use in the set membership problem. Both the theoretical advantages of SAT filters and experimental results show that this technique yields significant performance improvements over previous techniques. Specifically, a SAT filter is a filter construction that is simple yet efficient in terms of both query time and filter size; i.e., SAT filters asymptotically achieve the information-theoretic limit while providing fast querying. As well, this is the first application that makes use of the random k-SAT phase transition results and may drive research into efficient solvers for this and similar applications.

Stable Commutator Length in Free Products of Cyclic Groups

We give an algorithm to compute stable commutator length in free products of cyclic groups that is polynomial time in the length of the input, the number of factors, and the orders of the finite factors. We also describe some experimental and theoretical applications of this algorithm.

Stable immersions in orbifolds

We prove that in any hyperbolic orbifold with one boundary component, the product of any hyperbolic fundamental group element with a sufficiently large multiple of the boundary is represented by a geodesic loop that virtually bounds an immersed surface. In the case that the orbifold is a disk, there are some conditions. Our results generalize work of Calegari‐Louwsma and resolve a conjecture of Calegari. 20F65, 57M07; 57R42, 57R18