arXiv research

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…

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

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

Introduces a theorem for groups acting on trees.

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

Random quotients of hyperbolic cubulated groups remain cubulated.

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.

New findings on stable commutator lengths in recursively presented groups.

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

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

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…

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…

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.

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…

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 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 …

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,...,…

New anomaly cancellation formulas for E8*E8*E8 gauge group.

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

We prove that all hierarchically hyperbolic spaces have finite asymptotic dimension and obtain strong bounds on these dimensions. One application of this result is to obtain the sharpest known bound on the asymptotic dimension of the mapping class group of a finite type surface: improving the bound from exponential to …

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

We build quasi--isometry invariants of relatively hyperbolic groups which detect the hyperbolic parts of the group; these are variations of the stable dimension constructions previously introduced by the authors. We prove that, given any finite collection of finitely generated groups $\mathcal{H}$ each of which either …

Study infinite group presentations and their Dehn functions.

The paper extends elliptic genus and proves anomaly cancellation formulas for almost complex manifolds.

A remarkable result of Gersten states that the class of hyperbolic groups of cohomological dimension $2$ is closed under taking finitely presented (or more generally $FP_2$) subgroups. We prove the analogous result for relatively hyperbolic groups of Bredon cohomological dimension $2$ with respect to the family of para…

New Garside structures derived from groups, leading to new group properties.

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…

In this paper, we study the geometry of cone-offs of CAT(0) cube complexes over a family of combinatorially convex subcomplexes, with an emphasis on their Gromov-hyperbolicity. A first application gives a direct cubical proof of the characterization of the (strong) relative hyperbolicity of right-angled Coxeter groups,…

We give a lower and an upper bound for the conformal dimension of the boundaries of certain small cancellation groups. We apply these bounds to the few relator and density models for random groups. This gives generic bounds of the following form, where $l$ is the relator length, going to infinity. (a) $1 + 1/C < \Cdim(…

We study random walks on groups with the feature that, roughly speaking, successive positions of the walk tend to be "aligned". We formalize and quantify this property by means of the notion of deviation inequalities. We show that deviation inequalities have several consequences including Central Limit Theorems, the lo…

We study finite-dimensional representations of the Kauffman skein algebra of a surface S. In particular, we construct invariants of such irreducible representations when the underlying parameter q is a root of unity. The main one of these invariants is a point in the character variety consisting of group homomorphisms …

A group is coherent if all its finitely generated subgroups are finitely presented. In this article we provide a criterion for positively determining the coherence of a group. This criterion is based upon the notion of the perimeter of a map between two finite 2-complexes which is introduced here. In the groups to whic…

We introduce and study the notions of hyperbolically embedded and very rotating families of subgroups. The former notion can be thought of as a generalization of the peripheral structure of a relatively hyperbolic group, while the later one provides a natural framework for developing a geometric version of small cancel…

We introduce a local algorithm for Khovanov Homology computations - that is, we explain how it is possible to "cancel" terms in the Khovanov complex associated with a ("local") tangle, hence canceling the many associated "global" terms in one swoosh early on. This leads to a dramatic improvement in computational effici…

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.

We introduce the notion of directed diagrammatic reducibility which is a relative version of diagrammatic reducibility. Directed diagrammatic reducibility has strong group theoretic and topological consequences. A multi-relator version of the Freiheitssatz in the presence of directed diagrammatic reducibility is given.…

New results show metrics with positive scalar curvature can cancel on certain 4-manifolds.

In earlier work, Helen Wong and the author discovered certain "miraculous cancellations" for the quantum trace map connecting the Kauffman bracket skein algebra of a surface to its quantum Teichmueller space, occurring when the quantum parameter $q$ is a root of unity. The current paper is devoted to giving a more repr…

The Burau representation of braid group B4 is shown to be faithful almost everywhere.

Note on new cancellation formulas for manifolds.

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 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…

The paper conjectures and proves fixed points for certain group actions on nonpositively curved spaces.

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

In this paper, we derive an asymptotic formula for the number of conjugacy classes of elements in a class of statistically convex-cocompact actions with contracting elements. Denote by $\mathcal C(o, n)$ (resp. $\mathcal C'(o, n)$) the set of (resp. primitive) conjugacy classes of pointed length at most $n$ for a basep…

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…

We give a direct proof of a cancellation formula raised in [7] on the level of differential forms. We also obtain more cancellation formulas for even dimensional Riemannian manifolds with a complex line bundle involved. Relations among these cancellation formulas are discussed.