arXiv research

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

New spaces found without certain actions, using special subgroups.

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…

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…

Survey recent constructions of cyclic cocycles for Lie groups.

Given a proper, cocompact action of a Lie groupoid, we define a higher index pairing between invariant elliptic differential operators and smooth groupoid cohomology classes. We prove a cohomological index formula for this pairing by applying the van Est map and algebraic index theory. Finally we discuss in examples th…

Let $G$ be a group that admits a cocompact classifying space for proper actions $X$. We derive a formula for the Bredon cohomological dimension for proper actions of $G$ in terms of the relative cohomology with compact support of certain pairs of subcomplexes of $X$. We use this formula to compute the Bredon cohomologi…

Paradan and Vergne generalised the quantisation commutes with reduction principle of Guillemin and Sternberg from symplectic to Spin$^c$-manifolds. We extend their result to noncompact groups and manifolds. This leads to a result for cocompact actions, and a result for non-cocompact actions for reduction at zero. The r…

We introduce a Hopf algebroid associated to a proper Lie group action on a smooth manifold. We prove that the cyclic cohomology of this Hopf algebroid is equal to the de Rham cohomology of invariant differential forms. When the action is cocompact, we develop a generalized Hodge theory for the de Rham cohomology of inv…

We prove a Lichnerowicz type vanishing theorem for non-compact spin manifolds admiting proper cocompact actions. This extends a previous result of Ziran Liu who proves it for the case where the acting group is unimodular.

We show that the mapping class group of a closed surface admits a cocompact classifying space for proper actions of dimension equal to its virtual cohomological dimension.

Defines an equivariant index for proper actions by $H imes G$.

Sharpness of actions on reductive homogeneous spaces proven for various groups.

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

In this paper, we construct for the first time, the Witten genus and elliptic genera on noncompact manifolds with a proper cocompact action by an almost connected Lie group and prove vanishing and rigidity results that generalise known results for compact group actions on compact manifolds. We also compute our genera f…

Study shows mapping class group dimension for surfaces with punctures.

This paper studies the generic behavior of $k$-tuple elements for $k\ge 2$ in a proper group action with contracting elements, with applications towards relatively hyperbolic groups, CAT(0) groups and mapping class groups. For a class of statistically convex-cocompact action, we show that an exponential generic set of …

We give a superconnection proof of Connes' index theorem for proper cocompact actions of etale groupoids. This includes Connes' general foliation index theorem for foliations with Hausdorff holonomy groupoid.

Study index theory on Lie group homogeneous spaces using topological and analytic methods.

Analyzes semi-characteristics on specific manifolds, proving a vanishing theorem.

Study of groups acting on complex projective varieties.

The problem of equivariant rigidity is the $Γ$-homeomorphism classification of $Γ$-actions on manifolds with compact quotient and with contractible fixed sets for all finite subgroups of $Γ$. In other words, this is the classification of cocompact $E_{fin}Γ$-manifolds. We use surgery theory, algebraic $K$-theory, and t…

The paper introduces new topological obstructions for positive scalar curvature metrics on manifolds.

Solves a problem related to classifying spaces for proper actions and Nielsen Realization.

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 …

We study the index of the $G$-invariant elliptic pseudo-differential operator acting on a complete Riemannian manifold, where a unimodular, locally compact group $G$ acts properly and cocompactly. An $L^2$-index formula was obtained using the heat kernel method.

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

New groups act on cube complexes without compact cubulation.

Let G be a group and let M be a CAT(0) proper metric space (e.g. a simply connected complete Riemannian manifold of non-positive sectional curvature or a locally finite tree). Isometric actions of G on M are (by definition) points in the space R := Hom(G, Isom(M)) with the compact open topology. Sample theorems: 1. The…

The geometric dimension for proper actions $\underline{\mathrm{gd}}(G)$ of a group $G$ is the minimal dimension of a classifying space for proper actions $\underline{E}G$. We construct for every integer $r\geq 1$, an example of a virtually torsion-free Gromov-hyperbolic group $G$ such that for every group $Γ$ which con…

We show that most homogeneous Anosov actions of higher rank Abelian groups are locally smoothly rigid (up to an automorphism). This result is the main part in the proof of local smooth rigidity for two very different types of algebraic actions of irreducible lattices in higher rank semisimple Lie groups: (i) the Anosov…

Paper calculates indices for group actions using cocycles.

Let X be a proper CAT(0) cube complex admitting a proper cocompact action by a group G. We give three conditions on the action, any one of which ensures that X has a factor system in the sense of [BHS14]. We also prove that one of these conditions is necessary. This combines with results of Behrstock--Hagen--Sisto to s…

Wright showed that, if a 1-ended simply connected locally compact ANR Y with pro-monomorphic fundamental group at infinity admits a proper Z-action, then that fundamental group at infinity can be represented by an inverse sequence of finitely generated free groups. Geoghegan and Guilbault strengthened that result, prov…

Equivalence proven between equivariant K-theory and K-homology for certain matrix group actions.

Let $X$ be a proper CAT($0$) space and $G$ a cocompact group of isometries of $X$ without fixed point at infinity. We prove that if $\partial X$ contains an invariant subset of circumradius $π/2$, then $X$ contains a quasi-dense, closed convex subspace that splits as a product. Adding the assumption that the $G$-action…

We give a criterion in terms of the boundary for the existence of a proper cocompact action of a word-hyperbolic group on a CAT(0) cube complex. We describe applications towards lattices and hyperbolic 3-manifold groups. In particular, by combining the theory of special cube complexes, the surface subgroup result of Ka…

Guillemin trace formula adapted for group actions.

Characterizes convex cocompact actions in projective space with dynamical properties.

New criteria for non-isometric group actions in metric spaces.

The paper extends invariant theory to non-compact and non-reductive actions, classifying four regimes.

Study shows central limit theorem for counting measures in non-smooth spaces.

The energy of harmonic sections of flat bundles of nonpositively curved (NPC) length spaces over a Riemann surface $S$ is a function $E_ρ$ on Teichmüller space $\Teich$ which is a qualitative invariant of the holonomy representation $ρ$ of $π_1(S)$. Adapting ideas of Sacks-Uhlenbeck, Schoen-Yau and Tromba, we show that…

Let $X$ be a proper CAT(0) space and let $G$ be a cocompact group of isometries of $X$ which acts properly discontinuously. Charney and Sultan constructed a quasi-isometry invariant boundary for proper CAT(0) spaces which they called the contracting boundary. The contracting boundary imitates the Gromov boundary for $δ…

This paper explores topological aspects of index theory for infinite-dimensional manifolds.

We study proper, isometric actions of nonsolvable discrete groups Gamma on the 3-dimensional Minkowski space R^{2,1} as limits of actions on the 3-dimensional anti-de Sitter space AdS^3. To each such action is associated a deformation of a hyperbolic surface group Gamma_0 inside O(2,1). When Gamma_0 is convex cocompact…

Solves a problem related to Nielsen realization for certain groups.

We study actions of discrete subgroups $Γ$ of semi-simple Lie groups $G$ on associated oriented flag manifolds. These are quotients $G/P$, where the subgroup $P$ lies between a parabolic subgroup and its identity component. For Anosov subgroups $Γ\subset G$, we identify domains in oriented flag manifolds by removing a …