The Pure Virtual Braid Group Is Quadratic

If an augmented algebra K over Q is filtered by powers of its augmentation ideal I, the associated graded algebra grK need not in general be quadratic: although it is generated in degree 1, its relations may not be generated by homogeneous relations of degree 2. In this paper we give a sufficient criterion (called the PVH Criterion) for grK to be quadratic. When K is the group algebra of a group G, quadraticity is known to be equivalent to the existence of a (not necessarily homomorphic) universal finite type invariant for G. Thus the PVH Criterion also implies the existence of such a universal finite type invariant for the group G. We apply the PVH Criterion to the group algebra of the pure virtual braid group (also known as the quasi-triangular group), and show that the corresponding associated graded algebra is quadratic, and hence that these groups have a (not necessarily homomorphic) universal finite type invariant.Comment: 53 pages, 15 figures. Some clarifications added and inaccuracies corrected, reflecting suggestions made by the referee of the published version of the pape

Relation between two-phase quantum walks and the topological invariant

We study a position-dependent discrete-time quantum walk (QW) in one dimension, whose time-evolution operator is built up from two coin operators which are distinguished by phase factors from $x\geq0$ and $x\leq-1$. We call the QW the ${\it complete\;two}$-${\it phase\;QW}$ to discern from the two-phase QW with one defect[13,14]. Because of its localization properties, the two-phase QWs can be considered as an ideal mathematical model of topological insulators which are novel quantum states of matter characterized by topological invariants. Employing the complete two-phase QW, we present the stationary measure, and two kinds of limit theorems concerning ${\it localization}$ and the ${\it ballistic\;spreading}$, which are the characteristic behaviors in the long-time limit of discrete-time QWs in one dimension. As a consequence, we obtain the mathematical expression of the whole picture of the asymptotic behavior of the walker in the long-time limit. We also clarify relevant symmetries, which are essential for topological insulators, of the complete two-phase QW, and then derive the topological invariant. Having established both mathematical rigorous results and the topological invariant of the complete two-phase QW, we provide solid arguments to understand localization of QWs in term of topological invariant. Furthermore, by applying a concept of ${\it\;topological\;protections}$, we clarify that localization of the two-phase QW with one defect, studied in the previous work[13], can be related to localization of the complete two-phase QW under symmetry preserving perturbations.Comment: 50 pages, 13 figure

Groups of PL homeomorphisms of cubes

We study algebraic properties of groups of PL or smooth homeomorphisms of unit cubes in any dimension, fixed pointwise on the boundary, and more generally PL or smooth groups acting on manifolds and fixing pointwise a submanifold of codimension 1 (resp. codimension 2), and show that such groups are locally indicable (resp. circularly orderable). We also give many examples of interesting groups that can act, and discuss some other algebraic constraints that such groups must satisfy, including the fact that a group of PL homeomorphisms of the n-cube (fixed pointwise on the boundary) contains no elements that are more than exponentially distorted.Comment: 23 pages, 3 figure

Subshifts, MSO Logic, and Collapsing Hierarchies

We use monadic second-order logic to define two-dimensional subshifts, or sets of colorings of the infinite plane. We present a natural family of quantifier alternation hierarchies, and show that they all collapse to the third level. In particular, this solves an open problem of [Jeandel & Theyssier 2013]. The results are in stark contrast with picture languages, where such hierarchies are usually infinite.Comment: 12 pages, 5 figures. To appear in conference proceedings of TCS 2014, published by Springe