arXiv research

The aim of this paper is to study the spectrum of the $L^p$ Laplacian and the dynamics of the $L^p$ heat semigroup on non-compact locally symmetric spaces of higher rank. Our work here generalizes previously obtained results in the setting of locally symmetric spaces of rank one to higher rank spaces. Similarly as in t…

For real hyperbolic spaces, the dynamics of individual isometries and the geometry of the limit set of nonelementary discrete isometry groups have been studied in great detail. Most of the results were generalised to discrete isometry groups of simply connected Riemannian manifolds of pinched negative curvature. For sy…

Anosov representations give a higher-rank analogue of convex cocompactness in a rank-one Lie group which shares many of its good geometric and dynamical properties; geometric finiteness in rank one may be seen as a controlled weakening of convex cocompactness to allow for isolated failures of hyperbolicity. We introduc…

We study infinite covolume discrete subgroups of higher rank semisimple Lie groups, motivated by understanding basic properties of Anosov subgroups from various viewpoints (geometric, coarse geometric and dynamical). The class of Anosov subgroups constitutes a natural generalization of convex cocompact subgroups of ran…

This paper proves that there are no compact forms for a large class of homogeneous spaces admitting actions by higher-rank semisimple Lie groups. It builds on Zimmer's approach for studying such spaces using cocycle superrigidity. The proof involves cocycle superrigidity, measure rigidity for unipotent flows, technique…

Paper proves finite BMS measure for SPR groups in higher rank Lie groups.

The study proves no $L^2$-eigenvalues for higher rank locally symmetric spaces.

Defines new representations for hyperbolic groups, unifying existing definitions.

The paper proves rigidity for cocycles from higher rank lattices to Out(FN).

We study the geometry and dynamics of discrete infinite covolume subgroups of higher rank semisimple Lie groups. We introduce and prove the equivalence of several conditions, capturing "rank one behavior'' of discrete subgroups of higher rank Lie groups. They are direct generalizations of rank one equivalents to convex…

Affine maps reveal higher rank structures in certain spaces.

The paper encourages Kleinian group thinking for higher rank Lie groups.

Confirming a conjecture, new CAT(0) spaces of higher rank are rigid.

We prove two rigidity results for complete Riemannian three-manifolds of higher rank. Complete three-manifolds have higher spherical rank if an only if they are spherical space forms. Complete finite volume three-manifolds have higher hyperbolic rank if and only if they are hyperbolic space forms.

GeLoRA optimizes LoRA fine-tuning by dynamically adjusting ranks based on intrinsic dimensionality.

This survey is based on a series of lectures that we gave at MSRI in Spring 2015 and on a series of papers, mostly written jointly with Joan Porti. Our goal here is to: 1. Describe a class of discrete subgroups $Γ<G$ of higher rank semisimple Lie groups, which exhibit some "rank 1 behavior". 2. Give different character…

We prove that any action of a higher rank lattice on a Gromov-hyperbolic space is elementary. More precisely, it is either elliptic or parabolic. This is a large generalization of the fact that any action of a higher rank lattice on a tree has a fixed point. A consequence is that any quasi-action of a higher rank latti…

Study higher rank inner products and their tilings to describe tori degenerations.

Study critical exponents for L^p-cohomology of higher rank Lie groups and manifolds.

Characterizes higher rank model geometries using antipodal sets.

New proof for higher rank subvarieties in genus three.

New concept of coarse medians for higher rank symmetric spaces.

The study proves conditions for CAT(0) spaces with higher rank rigidity.

We show that polar actions of cohomogeneity two on simple compact Lie groups of higher rank, endowed with a biinvariant Riemannian metric, are hyperpolar. Combining this with a recent result of the second-named author, we are able to prove that polar actions induced by reductive algebraic subgroups in the isometry grou…

The paper proves actions of lattices in higher rank groups have cost one.

We give very flexible, concrete constructions of discrete and faithful epresentations of right-angled Artin groups into higher-rank Lie groups. Using the geometry of the associated symmetric spaces and the combinatorics of the groups, we find a general criterion for when discrete and faithful representations exist, and…

Improved homological dimension for certain subgroups in Lie groups.

Proposes a method to balance fairness and utility in ranking models.

The paper proves rigidity and ergodicity of horospherical foliations.

Unified view of spectral networks linking geometry and gauge theory.

New dHYM connections found on complex vector bundles.

For convex real projective manifolds we prove an analogue of the higher rank rigidity theorem of Ballmann and Burns-Spatzier.

In this paper, we introduce the notion of motif closure and describe higher-order ranking and link prediction methods based on the notion of closing higher-order network motifs. The methods are fast and efficient for real-time ranking and link prediction-based applications such as web search, online advertising, and re…

We show that sufficiently irreducible Anosov actions of higher rank abelian groups on tori and nilmanifolds are smoothly conjugate to affine actions.

Develops sublinear Morse theory in symmetric spaces.

Often, large, high dimensional datasets collected across multiple modalities can be organized as a higher order tensor. Low-rank tensor decomposition then arises as a powerful and widely used tool to discover simple low dimensional structures underlying such data. However, we currently lack a theoretical understanding …

The paper proves that certain spaces have injective balls of any radius.

We generalize the higher rank rigidity theorem to a class of Finsler spaces, i.e. Berwald spaces. More precisely, we prove that a complete connected Berwald space of finite volume and bounded nonpositive flag curvature with rank at least $2$ whose universal cover is irreducible, is a locally symmetric space or a locall…

Classifies measures for Anosov subgroups in higher ranks.

Enhances stock movement prediction using Higher Order Transformers for multimodal time-series data.

Researchers show how to perturb free group representations into higher rank groups.

A fast model estimates future prices from orderbook data.

We prove several cases of Zimmer's conjecture for actions of higher-rank cocompact lattices on low dimensional manifolds. For example, if $Γ$ is a cocompact lattice in $\mathrm{Sl}(n, \mathbb R)$, $M$ is a compact manifold, and $ω$ a volume form on $M$ we show that any homomorphism $ρ\colon Γ\rightarrow \mathrm{Diff}(M…

The study finds discrete subgroups with full limit sets in higher rank Lie groups.

Develops complex harmonic maps for Teichmüller theory, proving new theorems.

In this paper the notion of Measure Equivalence (ME) of countable groups is studied. ME was introduced by Gromov as a measure-theoretic analog of quasi-isometries. All lattices in the same locally compact group are Measure Equivalent; this is one of the motivations for this notion. The main result of this paper is ME r…

We classify all holomorphic actions of higher rank lattices on compact Kaehler manifolds of dimension 3. This provides a complete answer to Zimmer's program for holomorphic actions on compact Kaehler manifolds of dimension at most 3.

The prime geodesic theorem for regular geodesics in a higher rank locally symmetric space is proved. An application to class numbers is given. The proof relies on a Lefschetz formula that is based on work of Andreas Juhl.