arXiv research

The paper describes orbits of parabolic subgroups in complexified actions.

We consider automorphisms of homogeneous parabolic geometries with a fixed point. Parabolic geometries carry the distinguished distributions and we study those automorphisms which enjoy natural actions on the distributions at the fixed points. We describe the sets of such automorphisms on homogeneous parabolic geometri…

Abstract: Proves generic torus diffeomorphisms act parabolically and non-properly on fine curve graph and have generalized rotation sets.

The principal group of a Klein geometry has canonical left action on the homogeneous space of the geometry and this action induces action on the spaces of sections of vector bundles over the homogeneous space. This paper is about construction of differential operators invariant with respect to the induced action of the…

Groups' boundary actions are stable under small perturbations.

We study the Liouville action for quasi-Fuchsian groups with parabolic and elliptic elements. In particular, when the group is Fuchsian, the contribution of elliptic elements to the classical Liouville action is derived in terms of the Bloch-Wigner functions. We prove the first and second variation formulas for the cla…

The paper classifies symplectic invariants of specific singularities in integrable Hamiltonian systems.

Parabolic automorphisms on hyperkahler manifolds act ergodically on fibers.

We present an explicit construction of the moduli spaces of rank 2 stable parabolic bundles of parabolic degree 0 over the Riemann sphere, corresponding to "optimum" open weight chambers of parabolic weights in the weight polytope. The complexity of the different moduli space' weight chambers is understood in terms of …

Margulis space-times with parabolic holonomy elements are stable under sufficiently small deformations.

The moving coframe method is applied to solve the local equivalence problem for the class of linear parabolic equations in two independent variables under an action of the pseudo-group of contact transformations. The structure equations and the complete sets of differential invariants for symmetry groups are found. The…

In this paper we classify, up to orbit equivalence, cohomogeneity one actions of connected closed Lie subgroups of $U(1,n)$ on the $(2n+1)$-dimensional anti de Sitter spacetime $AdS^{2n+1}$. We also give some new examples of nonproper cohomogeneity one actions on $AdS^{n+1}$ and determine parabolic Lie subgroups of $SO…

The study connects monopole chains to Higgs bundles and classifies symmetric chains.

Study homeomorphisms on fine curve graph of surfaces, revealing new types of dynamics.

Let $\mathbf{E}$ be a flat Lorentzian space of signature $(2, 1)$. A Margulis space-time is a noncompact complete flat Lorentzian $3$-manifold $\mathbf{E}/Γ$ with a free holonomy group $Γ$ of rank $\mathbf{g}, \mathbf{g} \geq 2$. We consider the case when $Γ$ contains a parabolic element. We obtain a characterization o…

We define a torus action on the (complex) Cayley Grassmannian $X$. Using this action, we prove that $X$ is a singular variety. We also show that the singular locus is smooth and has the same cohomology ring as that of $\mathbb{CP}^5$. Furthermore, we identify the singular locus with a quotient of $G_2^\mathbb{C}$ by a …

We investigate the SL(2,R) invariant geodesic curves with the as- sociated invariant distance function in parabolic geometry. Parabolic geom- etry naturally occurs in the study of SL(2,R) and is placed in between the elliptic and the hyperbolic (also known as the Lobachevsky half-plane and 2- dimensional Minkowski half…

The study examines how perturbations of lattice actions on group boundaries behave.

A graph helps understand Artin groups better.

The paper consists of two parts. In the first one we show that a relatively hyperbolic group $G$ splits as a star graph of groups whose central vertex group is finitely generated and the other vertex groups are maximal parabolic subgroups. As a corollary we obtain that every group which admits 3-discontinuous and 2-coc…

Researchers compute the cohomology ring of a foliation defined by a group action.

Study slice-regular polynomial functions via twistor space group actions.

The aim of this paper is to study cohomogeneity one isometric linear actions on the $p+q$-dimensional pseudo-Euclidean space $\mathbb{R}^{p,q}$. It is proved that the natural isometric action of the nilpotent factor of an Iwasawa decomposition of $SO(p,q)$ is not of cohomogeneity one. The orbits of cohomogeneity one ac…

We consider the action of a finite subgroup of the mapping class group $Mod(S)$ of an oriented compact surface $S$ of genus $g \geq 2$ on the moduli space $\mathcal{R}(S,G)$ of representations of $π_1(S)$ in a connected semisimple real Lie group $G$. Kerckhoff's solution of the Nielsen realization problem ensures the e…

Study of parabolic Higgs bundles on curves with special fixed points.

Small deformations of a specific type of Lorentzian space-time preserve its structure.

We prove that the exceptional complex Lie group $F_4$ has a transitive action on the hyperplane section of the complex Cayley plane $\mathbb{OP}^2$. Our proof is direct and constructive. We use an explicit realization of the vector and spin actions of $\Spin(9,\C) \leq F_4$. Moreover, we identify the stabilizer of the …

The set of equivalence classes of cobounded actions of a group on different hyperbolic metric spaces carries a natural partial order. The resulting poset thus gives rise to a notion of the "best" hyperbolic action of a group as the largest element of this poset, if such an element exists. We call such an action a large…

The paper improves the description of Kähler metric flows and their singularities.

The paper classifies actions of a specific group on certain manifolds.

We prove that the Casimir operator acting on sections of a homogeneous vector bundle over a generalized flag manifold naturally extends to an invariant differential operator on arbitrary parabolic geometries. We study some properties of the resulting invariant operators and compute their action on various special types…

We prove that stability -- a strong quasiconvexity property -- pulls back under proper actions on proper metric spaces. This result has several applications, including that convex cocompact subgroups of both mapping class groups and outer automorphism groups of free groups are stable. We also characterize stability in …

Generalizing work of Haydys and Hitchin, we prove the existence of a hyperholomorphic line bundle on certain hyperkähler manifolds that do not necessarily admit an $S^1$ action. As examples, we consider the moduli space of (non-strongly) parabolic Higgs bundles, the moduli space of solutions to Nahm's equations, and Na…

We study the action of the elements of the mapping class group of a surface of finite type on the Teichmüller space of that surface equipped with Thurston's asymmetric metric. We classify such actions as elliptic, parabolic, hyperbolic and pseudo-hyperbolic, depending on whether the translation distance of such an elem…

Clarifies boundary criterion for non-one-ended subgroups in cubulation theory.

Non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.

The paper explores hidden torus symmetries in integrable systems and their stability.

Let $Λ_0$ be an ordered abelian group. We show how an $\mathrm{ATF}(\mathbb{Z}\timesΛ_0)$ group -- that is, a group admitting a free affine action without inversions on a $\mathbb{Z}\timesΛ_0$-tree -- admits a natural graph of groups decomposition, where vertex groups inherit actions on $Λ_0$-trees. Using recent work o…

We study here the action of subgroups of PSL(2,R) on the space of harmonic functions on the unit disc bounded by a common constant, as well as the relationship this action has with the foliated Liouville problem: Given a foliation of a compact manifold by Riemannian leaves and a leafwise harmonic continuous function on…

Donaldson defined a parabolic flow on Kahler manifolds which arises from considering the action of a group of symplectomorphisms on the space of smooth maps between manifolds. One can define a moment map for this action, and then consider the gradient flow of the square of its norm. Chen discovered the same flow from a…

Anosov groups in rank ≤3 have unique ergodic horospherical actions.

Defines connections on parabolic vector bundles for Lie algebroids.

Any action of a group $Γ$ on $\mathbb H^3$ by isometries yields a class in degree three bounded cohomology by pulling back the volume cocycle to $Γ$. We prove that the bounded cohomology of finitely generated Kleinian groups without parabolic elements distinguishes the asymptotic geometry of geometrically infinite ends…

Criterion found for Lie algebroid connections on parabolic bundles.

If f is a conformal mapping defined on a connected open subset of a Carnot group G, then either f is the composition of a translation, a dilation and an isometry, or G is the nilpotent Iwasawa component of a real rank 1 simple Lie group S, and f arises from the action of S on G, viewed as an open subset of S/P, where P…

Computes deformations of parabolic structures on Riemann surfaces.

Study of group boundaries and subgroup properties.

The study restricts matrix group actions on CAT(0) spaces and uniquely arcwise connected spaces, proving fixed points are inevitable.