arXiv research

Introduces Lie group actions in smoothing processes for currents and spaces with curvature.

Compact currents and charges in Carnot groups proved.

Current Lie groupoids generalize mappings to Lie groupoids.

This paper defines and studies currents and slices in the Heisenberg group, with new challenges and insights.

We study the properties of geodesic currents on free groups, particularly the "intersection form" that is similar to Bonahon's notion of the intersection number between geodesic currents on hyperbolic surfaces.

Currents on Lie groups form a Hopf algebra structure.

Subset currents on hyperbolic groups were introduced by Kapovich and Nagnibeda as a generalization of geodesic currents on hyperbolic groups, which were introduced by Bonahon and have been successfully studied in the case of the fundamental group $π_1 (Σ)$ of a compact hyperbolic surface $Σ$. Kapovich and Nagnibeda par…

We introduce and study the space of \emph{subset currents} on the free group $F_N$. A subset current on $F_N$ is a positive $F_N$-invariant locally finite Borel measure on the space $\mathfrak C_N$ of all closed subsets of $\partial F_N$ consisting of at least two points. While ordinary geodesic currents generalize con…

Maximal representations are studied using tree embeddings and geodesic currents.

Constructs currents and heights on K3 surfaces.

The paper proves existence and partial regularity for Legendrian area-minimizing currents.

We present an explicit realization of abelian extensions of infinite dimensional Lie groups using abelian extensions of path groups, by generalizing Mickelsson's approach to loop groups and the approach of Losev-Moore-Nekrasov-Shatashvili to current groups. We apply our method to coupled cocycles on current Lie algebra…

We show how the fundamental cocycles on current Lie algebras and the Lie algebra of symmetries for the sigma model are obtained via the current algebra functors. We present current group extensions integrating some of these current Lie algebra extensions.

Study automorphism groups of geodesic currents and measured laminations on surfaces.

New metric on geodesic currents connects different surface genera.

The paper studies automorphisms of free groups with a North-South dynamics.

A projection maps geodesic currents to Teichmüller space.

New space of currents defined for nonabelian free groups and malnormal subgroups.

New characterization of geodesic currents via curve functionals.

The paper proves Rademacher's theorem for Heisenberg groups.

Geodesic currents on hyperbolic surfaces have dual spaces that are metric trees.

Extends curve functions to geodesic currents with a simple criterion.

Let $\varphi$ be a hyperbolic outer automorphism of a non-abelian free group $F_N$ such that $\varphi$ and $\varphi^{-1}$ admit absolute train track representatives. We prove that $\varphi$ acts on the space of projectivized geodesic currents on $F_N$ with generalized uniform North-South dynamics.

We examine the theory of metric currents of Ambrosio and Kirchheim in the setting of spaces admitting differentiable structures in the sense of Cheeger and Keith. We prove that metric forms which vanish in the sense of Cheeger on a set must also vanish when paired with currents concentrated along that set. From this we…

New group not biautomatic, geometrically constructed.

Let (ρ_λ)_{λ\in Λ} be a holomorphic family of representations of a finitely generated group G into PSL(2,C), parameterized by a complex manifold Λ. We define a notion of bifurcation current in this context, that is, a positive closed current on Λdescribing the bifurcations of this family of representations in a quantit…

Study geodesic currents on surfaces, proving new properties of character varieties.

Currents with corners help count triangulations on surfaces.

We obtain formulas for the first and second cohomology groups of a general current Lie algebra with coefficients in the "current" module, and apply them to compute structure functions for manifolds of loops with values in compact Hermitian symmetric spaces.

Let (ρ_\la)_{\la\in \La} be a holomorphic family of representations of a surface group π_1(S) into PSL(2,C), where S is a topological (possibly punctured) surface with negative Euler characteristic. Given a structure of Riemann surface of finite type on S we construct a bifurcation current on the parameter space \La, t…

We study the structure of abelian extensions of the group $L_qG$ of $q$-differentiable loops (in the Sobolev sense), generalizing from the case of central extension of the smooth loop group. This is motivated by the aim of understanding the problems with current algebras in higher dimensions. Highest weight modules are…

Geodesics and boundaries found for metric structures on hyperbolic groups.

We quantitatively relate the Patterson-Sullivant currents and generic stretching factors for free group automorphisms to the asymmetric Lipschitz metric on Outer space and to Guirardel's intersection number.

Let $F_N$ be a free group of rank $N\ge 2$, let $μ$ be a geodesic current on $F_N$ and let $T$ be an $\mathbb R$-tree with a very small isometric action of $F_N$. We prove that the geometric intersection number $<T, μ>$ is equal to zero if and only if the support of $μ$ is contained in the dual algebraic lamination $L^…

We find a canonical decomposition of a geodesic current on a surface of finite type arising from a topological decomposition of the surface along special geodesics. We show that each component either is associated to a measured lamination or has positive systole. For a current with positive systole, we show that the in…

New insights into currents of Hitchin representations with combinatorial restrictions.

Study shows negatively curved manifolds' spherical volume equals minimal surface area.

A \emph{geodesic current} on a free group $F$ is an $F$-invariant measure on the set $\partial^2 F$ of pairs of distinct points of $\partial F$. The space of geodesic currents on $F$ is a natural companion of Culler-Vogtmann's Outer space $cv(F)$ and studying them together yields new information about both spaces as we…

We expose a K-theoretic approach to study group C*-algebras and C*-algebraic compact quantum groups: 1. The conception of multidimensional geometric quantization and the index of group C*-algebras; 2. the entire homology of noncommutative de Rham currents and the noncommutative Chern characters, and their computation f…

Geometric correspondence links flow metrics to reparameterizations.

Let $S$ be a compact, connected, oriented surface, possibly with boundary, of negative Euler characteristic. In this article we extend Lindenstrauss-Mirzakhani's and Hamenstädt's classification of locally finite mapping class group invariant ergodic measures on the space of measured laminations $\mathcal{M}\mathcal{L}(…

We compare the homology groups $H_n ^{IC}(X)$ of the chain complex of integral currents with compact support of a metric space $X$ with the singular Lipschitz homology $H^L_n (X)$ and with ordinary singular homology. If $X$ satisfies certain cone inequalities all these homology theories coincide. On the other hand, for…

This thesis treats two main topics: calibrated symplectic foliations, and local Lie groupoids. Calibrated symplectic foliations are one possible generalization of taut foliations of 3-manifolds to higher dimensions. Their study has been popular in recent years, and we collect several interesting results. We then show h…

This article will explore the K- and L-theory of group rings and their applications to algebra, geometry and topology. The Farrell-Jones Conjecture characterizes K- and L-theory groups. It has many implications, including the Borel and Novikov Conjectures for topological rigidity. Its current status, and many of its co…

Expanding on a Heisenberg group case for a mathematical proposition.

The paper examines convergence of currents and forms under smooth diffeomorphisms.

We show that all the currently known non-arithmetic lattices in ${\rm PU}(2,1)$ are monodromy groups of higher hypergeometric functions.

WZW models are abstract conformal field theories with an infinite dimensional symmetry which accounts for their integrability, and at the same time they have a sigma model description of closed string propagation on group manifolds which, in turn, endows the models with an intuitive geometric meaning. We exploit this d…