arXiv research

Using small cancellation for rotating families of groups, we construct new examples of aspherical polyhedra.

Random quotients of hyperbolic cubulated groups remain cubulated.

In a pair of recent papers (one to appear and one forthcoming), the author develops a general version of small cancellation theory applicable in higher dimensions, and then applies this theory to the Burnside groups of sufficiently large exponent. The present article gives a brief introduction to the methods and techni…

New proof shows Cohen-Lyndon property for non-metric small-cancellation.

The paper introduces new invariants to detect hyperbolic parts in relatively hyperbolic groups.

This thesis consists of three self-contained chapters. The first two concern quantum invariants of links and three manifolds and the third contains results on the word problem for link groups. In chapter 1 we relate the tree part of the Aarhus integral to the mu-invariants of string-links in homology balls thus general…

Random branched covers of groups are homotopy equivalent to geometrically small cancellation complexes.

Introduces a theorem for groups acting on trees.

We explain and generalise a construction due to Gromov to realise geometric small cancellation groups over graphs of groups as fundamental groups of non-positively curved 2-dimensional complexes of groups. We then give conditions so that the hyperbolicity and some finiteness properties of the small cancellation quotien…

Proves small cancellation free products have geometric actions on CAT(0) cube complexes.

We use a simple geometric argument and small cancellation properties of link groups to prove that alternating links are non-trivial. This proof uses only classic results in topology and combinatorial group theory.

We prove that a group obtained as a quotient of the free product of finitely many cubulable groups by a finite set of relators satisfying the classical $C'(1/6)$--small cancellation condition is cubulable. This yields a new large class of relatively hyperbolic groups that can be cubulated, and constitutes the first ins…

Cohomotopy theory predicts M-theory anomaly cancellation on 8-manifolds.

We give an alternate proof of Wise's Malnormal Special Quotient Theorem (MSQT), avoiding cubical small cancellation theory. We also show how to deduce Wise's Quasiconvex Hierarchy Theorem from the MSQT and theorems of Hsu--Wise and Haglund--Wise.

Introduces directed diagrammatic reducibility with group and topological implications.

CR singularities in 3-manifolds can be cancelled by an isotopy supported in an arbitrarily small neighborhood of a Seifert surface.

New framework solves string theory's RR-field tadpole cancellation problems.

Resolves anomaly cancellation for M5-brane in M-theory.

We study the geometry of infinitely presented groups satisfying the small cancelation condition C'(1/8), and define a standard decomposition (called the criss-cross decomposition) for the elements of such groups. We use it to prove the Rapid Decay property for groups with the stronger small cancelation property C'(1/10…

New group with non-loxodromic Morse element found.

Abstract: Generalizes modular forms to family case and finds new anomaly cancellation formulas.

The paper proves spaces associated to certain cubical presentations are aspherical.

New findings on stable commutator lengths in recursively presented groups.

Study simplicial volume and stable commutator length for one-relator groups.

Holomorphic supergravity theory simplifies anomaly cancellation in heterotic moduli.

Thurston conjectured that a closed triangulated 3-manifold in which every edge has degree 5 or 6, and no two edges of degree 5 lie in a common 2-cell, has word-hyperbolic fundamental group. We establish Thurston's conjecture by proving that such a manifold admits a piecewise Euclidean metric of non-positive curvature a…

String structures in degree four are associated with cancellation of anomalies of string theory in ten dimensions. Fivebrane structures in degree eight have recently been shown to be associated with cancellation of anomalies associated to the NS5-brane in string theory as well as the M5-brane in M-theory. We introduce …

The paper examines cone-offs of CAT(0) cube complexes and their geometric properties.

Review of non-abelian gerbes and their applications in string theory.

The main new result here is the cancellation of global anomalies in the Type I superstring, with and without D-branes. Our argument here depends on a precise interpretation of the 2-form abelian gauge field using KO-theory; then the anomaly cancellation follows from a geometric form of the full Atiyah-Singer index theo…

Study on non-peripheral ideal decompositions of alternating knots.

Finite asymptotic dimension proved for hierarchically hyperbolic spaces.

We show that a group with a presentation satisfying the C(6) small cancellation condition cannot contain a subgroup isomorphic to F_2 X F_2.

Study of massless spectrum in heterotic compactifications.

Study infinite group presentations and their Dehn functions.

Proves M-theory anomaly cancellation on nonorientable manifolds.

We study the cohomological physics of fivebranes in type II and heterotic string theory. We give an interpretation of the one-loop term in type IIA, which involves the first and second Pontrjagin classes of spacetime, in terms of obstructions to having bundles with certain structure groups. Using a generalization of th…

It has been shown that the Alvarez-Gaum$\mathrm{\acute{e}}$-Witten miraculous anomaly cancellation formula in type IIB superstring theory and its various generalizations can be derived from modularity of certain characteristic forms. In this paper, we show that the Green-Schwarz formula and the Schwarz-Witten formula i…

Note on new cancellation formulas for manifolds.

Anomaly cancellation condition for M5-brane action is proven using twisted Cohomotopy.

The order submission and cancelation processes are two crucial aspects in the price formation of stocks traded in order-driven markets. We investigate the dynamics of order cancelation by studying the statistical properties of inter-cancelation durations defined as the waiting times between consecutive order cancelatio…

Let $G$ be a group given by the presentation [<a_1,...,a_k,b_1,... b_k\,| a_i=u_i(\bar b), b_i=v_i(\bar a) \hbox{for} 1\le i\le k>,] where $k\ge 2$ and where the $u_i\in F(b_1,..., b_k)$ and $w_i\in F(a_1,..., a_k)$ are random words. Generically such a group is a small cancellation group and it is clear that $(a_1,...,…

Study modular forms over Γ^0(2) and anomaly cancellation formulas.

We analyze global anomalies for elementary Type II strings in the presence of D-branes. Global anomaly cancellation gives a restriction on the D-brane topology. This restriction makes possible the interpretation of D-brane charge as an element of K-theory.

We give a precise and concise formulation of the orientifold construction in Type II superstring theory. Our results include anomaly cancellation on the worldsheet and a spacetime computation of the background Ramond-Ramond charge.

Anomaly flow uses superstring theory to maintain metric properties.

We find new bounds on the conformal dimension of small cancellation groups. These are used to show that a random few relator group has conformal dimension 2+o(1) asymptotically almost surely (a.a.s.). In fact, if the number of relators grows like l^K in the length l of the relators, then a.a.s. such a random group has …

Order cancellation process plays a crucial role in the dynamics of price formation in order-driven stock markets and is important in the construction and validation of computational finance models. Based on the order flow data of 18 liquid stocks traded on the Shenzhen Stock Exchange in 2003, we investigate the empiric…