arXiv research

New method constructs proper affine actions of groups in higher dimensions.

Proves nonemptyness of domains for specific group actions.

The paper generalizes deformation results for Fuchsian representations and shows proper affine actions.

Discrete groups act properly on 3-space, solving Milnor's question.

The paper defines cocycles for positive Anosov representations and constructs affine actions with bounded fundamental domains.

For any right-angled Coxeter group $Γ$ on $k$ generators, we construct proper actions of $Γ$ on $\mathrm{O}(p,q+1)$ by right and left multiplication, and on the Lie algebra $\mathfrak{o}(p,q+1)$ by affine transformations, for some $p,q\in\mathbb N$ with $p+q+1=k$. As a consequence, any virtually special group admits pr…

We define a fuchsian affine action of a surface group to be such that the linear part factors through a representation of $SL(2,{\mathbb R})$. We prove a fuchsian affine action of a surface group is never proper.

New insights into surface group actions and entropy.

We give necessary and sufficient conditions for an affine deformation of a Schottky subgroup of O(2,1) to act properly on affine space. There exists a real-valued biaffine map between the cohomology of the Schottky group and the space of geodesic currents on the corresponding hyperbolic surface S. For a fixed cohomolog…

We show that for any group $G$ that is hyperbolic relative to subgroups that admit a proper affine isometric action on a uniformly convex Banach space, then $G$ acts properly on a uniformly convex Banach space as well.

A Margulis spacetime is a complete flat affine Lorentzian 3-manifold with free fundamental group. Associated to $M$ is a noncompact complete hyperbolic surface $Σ$. We study double extensions of $π_1 (M) \cong π_1 (Σ)$ when $Σ$ is homeomorphic to a projective plane minus two discs. We classify proper actions of this do…

We will first clarify the loop group formulations for both hyperbolic and elliptic definite affine spheres in R^3. Then we classify the rational elements with 3 poles or 6 poles in a real twisted loop group, and compute dressing actions of them on such surfaces. Some new examples with pictures will be produced at last.

We consider the deformation of a discontinuous group acting on the Euclidean space by affine transformations. A distinguished feature here is that even a `small' deformation of a discrete subgroup may destroy proper discontinuity of its action. In order to understand the local structure of the deformation space of disc…

Polynomial Duistermaat-Heckman measure on symplectic groupoid quotients.

We define the notion of affine Anosov representations of word hyperbolic groups into the affine group $\mathsf{SO}^0(n+1,n)\ltimes\mathbb{R}^{2n+1}$. We then show that a representation $ρ$ of a word hyperbolic group is affine Anosov if and only if its linear part $\mathtt{L}_ρ$ is Anosov in $\mathsf{SO}^0(n+1,n)$ with …

The action dimension of a discrete group $G$ is the minimum dimension of contractible manifold that admits a proper $G$-action. We compute the action dimension of the direct limit of a simple complex of groups for several classes of examples including: 1) Artin groups, 2) graph products of groups, and 3) fundamental gr…

We study the possibility of applying a finite-dimensionality argument in order to address parts of the Baum-Connes conjecture for finitely generated linear groups. This gives an alternative approach to the results of Guentner, Higson, and Weinberger concerning the Baum-Connes conjecture for linear groups. For any finit…

We classify the simplest rational elements in a twisted loop group, and prove that dressing actions of them on proper indefinite affine spheres give the classical Tzitzéica transformation and its dual. We also give the group point of view of the Permutability Theorem, construct complex Tzitzéica transformations, and di…

This paper classifies Hamiltonian actions by symplectic groupoids using Delzant subspaces.

A study of proper affine vector fields in plane symmetric static space-times by using the rank of the Rieman matrix and holonomy. Studying proper affine vector fields in each case, It is shown that the special class of the above space-times admit proper affine vector fields.

Extends PF submanifold results and connects Kac-Moody spaces.

The paper extends a theorem about momentum maps to singular symplectic spaces.

In this article, we provide a necessary and sufficient criterion for proper actions on $\mathbb{H}^{n,n-1}$ in terms of certain special Anosov representations in $\mathsf{SO}(n,n)$. Moreover, we show that affine Anosov representations of any word hyperbolic group in $\mathsf{SO}(n,n-1)\ltimes\mathbb{R}^{2n-1}$ are infi…

Study on descent properties of complex affine surfaces under proper morphisms.

Proper proximality proved for various groups on non-positive curvature spaces.

Study boundary actions on CAT(0) spaces, proving topological freeness.

We construct finite volume hyperbolic manifolds with large symmetry groups. The construction makes use of the presentations of finite Coxeter groups provided by Barot and Marsh and involves mutations of quivers and diagrams defined in the theory of cluster algebras. We generalize our construction by assigning to every …

We show that the natural S^1-bundle over a projective special Kaehler manifold carries the geometry of a proper affine hypersphere endowed with a Sasakian structure. The construction generalizes the geometry of the Hopf-fibration $\Sr^{2n+1} \longrightarrow \CP^n$ in the context of projective special Kaehler manifolds.…

We consider non-degenerate centro-affine hypersurface immersions in R^n whose cubic form is parallel with respect to the Levi-Civita connection of the affine metric. There exists a bijective correspondence between homothetic families of proper affine hyperspheres with center in the origin and with parallel cubic form, …

For every proper convex cone $K \subset \mathbb R^3$ there exists a unique complete hyperbolic affine 2-sphere with mean curvature $-1$ which is asymptotic to the boundary of the cone. Two cones are associated if the corresponding affine spheres can be mapped to each other by an orientation-preserving isometry. This eq…

A generalized semitoric system F:=(J,H): M --> R^2 on a symplectic 4-manifold is an integrable system whose essential properties are that F is a proper map, its set of regular values is connected, J generates an S^1-action and is not necessarily proper. These systems can exhibit focus-focus singularities, which corresp…

Non-proper surface group action on product of trees found.

Generalizing the notion of domains of dependence in the Minkowski space, we define and study regular domains in the affine space with respect to a proper convex cone. In dimension three, we show that every proper regular domain is uniquely foliated by a particular kind of surfaces with constant affine Gaussian curvatur…

Proper actions on bornological spaces are characterized with compatible coarse structures.

In his 1990 doctoral thesis, Todd Drumm showed that proper affine deformations of free Fuchsian groups could be constructed as Schottky groups using a new family of hypersurfaces called "crooked planes." The existence of proper affine deformations of Fuchsian Schottky groups was demonstrated by Margulis in the early 19…

The paper explores proper actions and their relation to representation theory, with new quantitative methods.

Reduces proper actions to simpler core actions for analysis.

No policy can simultaneously be fully autonomous, optimally calibrated, and helpful, proving a trilemma.

Study proper actions of Lie groups on symmetric spaces, finding rigidity results and Hurwitz-Radon numbers.

New spaces found without certain actions, using special subgroups.

Reduction principles for proper actions on smooth manifolds.

Given a group action on a simplicial complex such that each simplex stabiliser admits a cocompact model of classifying space for proper actions, we give conditions implying the existence of a cocompact model of classifying space for proper actions for the whole group. This is used to generalise previous combination res…

We establish a Lichnerowicz type vanishing theorem for non-compact spin manifolds admiting proper cocompact actions, when the action group is unimodular.

We first introduce an invariant index for G-equivariant elliptic differential operators on a locally compact manifold M admitting a proper cocompact action of a locally compact group G. It generalizes the Kawasaki index for orbifolds to the case of proper cocompact actions. Our invariant index is used to show that an a…

The paper explores conformal groups on plane waves and proves a conjecture in locally homogeneous settings.

In his seminal work \cite{pal:61}, R. Palais extended a substantial part of the theory of compact transformation groups to the case of proper actions of locally compact groups. Here we extend to proper actions some other important results well known for compact group actions. In particular, we prove that if $H$ is a co…

A theory of double affine and special double affine bundles, i.e. differential manifolds with two compatible (special) affine bundle structures, is developed as an affine counterpart of the theory of double vector bundles. The motivation and basic examples come from Analytical Mechanics, where double affine bundles hav…

We construct homogeneous flat pseudo-Riemannian manifolds with non-abelian fundamental group. In the compact case, all homogeneous flat pseudo-Riemannian manifolds are complete and have abelian linear holonomy group. To the contrary, we show that there do exist non-compact and non-complete examples, where the linear ho…