arXiv research

In this technical note we give a purely geometric understanding of discrete torsion, as an analogue of orbifold Wilson lines for two-form tensor field potentials. In order to introduce discrete torsion in this context, we describe gerbes and the description of certain type II supergravity tensor field potentials as con…

In this paper, we consider the discrete deformation of the discrete space curves with constant torsion described by the discrete mKdV or the discrete sine-Gordon equations, and show that it is formulated as the torsion-preserving equidistant deformation on the osculating plane which satisfies the isoperimetric conditio…

Study curvature and torsion from cross-ratios in discrete curves.

Analytic torsion defined for non-compact Lie groups and discrete subgroups.

In this article, we study the twisting procedure of orbifold cohomology. We introduce local system and construct twisted orbifold cohomology. Then, we generalize Vafa-Witten's notion of discrete torsion to general orbifold and examine its relation to local system. Finally, some examples are computed.

Any two equivalent discrete curves must have the same invariants at the corresponding points under an affine transformation. In this paper, we construct the moving frame and invariants for the discrete centroaffine curves, which could be used to discriminate the same discrete curves from different graphics, and estimat…

We consider the interpretation in classical geometry of conformal field theories constructed from orbifolds with discrete torsion. In examples we can analyze, these spacetimes contain ``stringy regions'' that from a classical point of view are singularities that are to be neither resolved nor blown up. Some of these mo…

We characterize finitely generated torsion-free Kleinian groups for which the real length spectrum (without multiplicities) is discrete.

We generalize Bonahon's characterization of geometrically infinite torsion-free discrete subgroups of PSL(2, $\mathbb{C}$) to geometrically infinite discrete isometry subgroups in the case of rank 1 symmetric spaces, and, under the assumption of bounded torsion, to the case of negatively pinched Hadamard manifolds. Eve…

This is the writeup of a lecture given at the May Wisconsin workshop on mathematical aspects of orbifold string theory. In the first part of this lecture, we review recent work on discrete torsion, and outline how it is currently understood in terms of the B field. In the second part of this lecture, we discuss the rel…

Paper introduces new $L_p$ $q$-torsional measure and solves Minkowski problem.

Discretization of curves is an ancient topic. Even discretization of curves with an eye toward differential geometry is over a century old. However there is no general theory or methodology in the literature, despite the ubiquitous use of discrete curves in mathematics and science. There are conflicting definitions of …

In a previous paper we outlined how discrete torsion can be understood geometrically as an analogue of orbifold U(1) Wilson lines. In this paper we shall prove the remaining details. More precisely, in this paper we describe gerbes in terms of objects known as stacks (essentially, sheaves of categories), and develop mu…

Graphically discrete groups have strong rigidity properties.

Let $H^n$ be the hyperbolic n-space with $n\geq 2$. Suppose that $Γ<Isom H^n$ is a discrete, torsion free subgroup and $a$ is a point in the domain of discontinuity $Ω(Γ)$. Let $p$ be the projection map from $H^n$ to the quotient manifold $M=H^n/Γ$. In this paper we prove that there exists an open neighborhood $U$ of $…

In this paper we introduce the concept of Deligne cohomology of an orbifold. We prove that the third Deligne cohomology group of a smooth étale groupoid classify gerbes with connection over the groupoid. We argue that the $B$-field and the discrete torsion in type II superstring theories are special kinds of gerbes wit…

New polynomial connects knot genus to 3-manifold geometry.

Discrete Morse theory simplifies Khovanov homology calculations.

The paper studies fundamental groups of orbit configuration spaces and proves their torsion-freeness.

In this paper, we generalize Bonahon's characterization of geometrically infinite torsion-free discrete subgroups of PSL(2, $\mathbb{C}$) to geometrically infinite discrete subgroups $Γ$ of isometries of negatively pinched Hadamard manifolds $X$. We then generalize a theorem of Bishop to prove that every discrete geome…

Improved homological dimension for certain subgroups in Lie groups.

Given a fiber bundle $Z \to M \to B$ and a flat vector bundle $E \to M$ with a compatible action of a discrete group $G$, and regarding $B / G$ as the non-commutative space corresponding to the crossed product algebra, we construct an analytic torsion form as a non-commutative deRham differential form. We show that our…

The paper connects geodesic flows and higher-dimensional Reidemeister torsion for hyperbolic orbifolds.

Study of a torsion class in mapping class group's cohomology.

A linkage mechanism consists of rigid bodies assembled by joints which can be used to translate and transfer motion from one form in one place to another. In this paper, we are particularly interested in a family of spacial linkage mechanisms which consist of $n$-copies of a rigid body joined together by hinges to form…

If $M$ is a finite volume complete hyperbolic 3-manifold with one cusp and no 2-torsion, the geometric component $X_M$ of its $\SL(2,\BC)$-character variety is an affine complex curve, which is smooth at the discrete faithful representation $ρ_0$. Porti defined a non-abelian Reidemeister torsion in a neighborhood of $ρ…

We give a quantum field theoretic derivation of the formula obeyed by the Ray-Singer torsion on product manifolds. Such a derivation has proved elusive up to now. We use a BRST formalism which introduces the idea of an infinite dimensional Universal Gauge Fermion, and is of independent interest being applicable to situ…

The study proves a discrete version of Segre's theorem for polygonal curves.

We study the existence of invariant metrics with holonomy $G_{2(2)}^* \subset SO(4,3)$ on compact nilmanifolds, i.e. on compact quotients of nilpotent Lie groups by discrete subgroups. We prove that, up to isomorphism, there exists only one indecomposable nilpotent Lie algebra admitting a torsion-free $G_{2(2)}^*$-stru…

Simplicial versions of topological abelian gauge theories are constructed which reproduce the continuum expressions for the partition function and Wilson expectation value of linked loops, expressible in terms of R-torsion and linking numbers respectively. The new feature which makes this possible is the introduction o…

Extends Nielsen realization to infinite-type surfaces, classifying torsion elements and topological groups.

Study continuous paths in discrete subgroups of hyperbolic space, proving combination and decomposition theorems.

In this paper we get an explicit lower bound for the radius of a Bergman ball contained in the Dirichlet fundamental polyhedron of a torsion-free discrete group $G\subset PU(n,1)$ acting on complex hyperbolic space. Consequently the volume of all complex hyperbolic n-manifolds is bounded below by the volume of this bal…

Given an almost complex manifold (M, J), we study complex connections with trivial holonomy and such that the corresponding torsion is either of type (2,0) or of type (1,1) with respect to J. Such connections arise naturally when considering Lie groups, and quotients by discrete subgroups, equipped with bi-invariant an…

Study asymptotic expansion of graph Laplacian on discretized surfaces, relating spanning trees and cycle-rooted forests.

For any discrete, torsion-free subgroup $Γ$ of $\mathrm{Sp}(n,1)$ (resp.\ $\mathrm{F}_4^{-20}$) with no parabolic elements, we prove that $H_{4n-1}(Γ;V)=0$ (resp.\ $H_i(Γ;V)=0$ for $i=13,14,15$) for any $Γ$--module $V$. The main technical advance is a new bound on the $p$--Jacobian of the barycenter map of Besson--Cour…

In the current article our primary objects of study are compact complex submanifolds of quotient manifolds of irreducible bounded symmetric domains by torsion free discrete lattices of automorphisms. We are interested in the characterization of the totally geodesic submanifolds among compact splitting complex submanifo…

We propose a new method for studying $n$- and $Γ$-cohomology of globalizations of Harish-Chandra modules, where $G=KAN$ is a rank one semisimple Lie group, $Γ$ is a discrete subgroup of $G$ and $n=Lie(N)$. We prove a conjecture of Patterson relating the singularities of Selberg zeta functions with the $Γ$-cohomology of…

Group lattices (Cayley digraphs) of a discrete group are in natural correspondence with differential calculi on the group. On such a differential calculus geometric structures can be introduced following general recipes of noncommutative differential geometry. Despite of the non-commutativity between functions and (gen…

Extends Lannes-Quillen theorem to all profinite groups.

Study eigenvalues and functions on specific Heisenberg manifolds.

We prove that any countable discrete and torsion free subgroup of a general linear group over an arbitrary field or a similar subgroup of an almost connected Lie group satisfies the integral algebraic K-theoretic (split) Novikov conjecture over \cpt and §, where \cpt denotes the C^*-algebra of compact operators and §de…

We introduce the notion of volume of the representation variety of a finitely presented discrete group in a compact Lie group using the push-forward measure associated to a map defined by a presentation of the discrete group. We show that the volume thus defined is invariant under the Andrews-Curtis moves of the genera…

For hyperbolic Riemann surfaces of finite geometry, we study Selberg's zeta function and its relation to the relative scattering phase and the resonances of the Laplacian. As an application we show that the conjugacy class of a finitely generated, torsion-free, discrete subgroup of SL(2,R) is determined by its trace sp…

Paper proves equivariant Fried conjecture for specific flows.

Complex hyperbolic Kleinian groups yield Stein manifolds under certain conditions.

The paper analyzes spectral properties of connection Laplacian on tori, proving convergence to real torus.

Groups with certain properties have invariant subalgebra rigidity.