arXiv math/05042412005-04-12math.GRmath.DGmath.MG

Superrigidity for irreducible lattices and geometric splitting

Nicolas Monod

备注：: Improved version of earlier preprint. Definitions 3, 5 and proof of Theorem 55 modified
发表信息：: See final version in: J. AMS 19 No. 4 (2006) 781--814
DOI：: 10.1090/S0894-0347-06-00525-X

英文摘要：

We prove general superrigidity results for actions of irreducible lattices on CAT(0) spaces; first, in terms of the ideal boundary, and then for the intrinsic geometry (including for infinite-dimensional spaces). In particular, one obtains a new and self-contained proof of Margulis' superrigidity theorem for uniform irreducible lattices in non-simple groups. The proofs rely on simple geometric arguments, including a splitting theorem which can be viewed as an infinite-dimensional (and singular) generalization of the Lawson-Yau/Gromoll-Wolf theorem.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Cocycle superrigidity for profinite actions of irreducible lattices

Let $Γ$ be an irreducible lattice in a product of two locally compact groups and assume that $Γ$ is densely embedded in a profinite group $K$. We give necessary conditions which imply that the left translation action $Γ\curvearrowright K$ is "virtually" cocycle superrigid: any cocycle $w:Γ\times K\rightarrowΔ$ with values in a countable group $Δ$ is cohomologous to a cocycle which factors through the map $Γ\times K\rightarrowΓ\times K_0$, for some finite quotient group $K_0$ of $K$. As a corollary, we deduce that any ergodic profinite action of $Γ=\text{SL}_2(\mathbb Z[S^{-1}])$ is virtually c…

2021-03-30math.DSmath.GRmath.OA

Arithmeticity vs. non-linearity for irreducible lattices

We establish an arithmeticity vs. non-linearity alternative for irreducible lattices in suitable product groups, such as for instance products of topologically simple groups. This applies notably to a (large class of) Kac-Moody groups. The alternative relies on a CAT(0) superrigidity theorem, as we follow Margulis' reduction of arithmeticity to superrigidity.

2004-07-06math.GR

Superrigidity, generalized harmonic maps and uniformly convex spaces

We prove several superrigidity results for isometric actions on metric spaces satisfying some convexity properties. First, we extend some recent theorems of N. Monod on uniform and certain non-uniform irreducible lattices in products of locally compact groups. Second, we include the proof of an unpublished result on commensurability superrigidity due to Margulis. The proofs rely on certain notions of harmonic maps and the study of their existence, uniqueness, and continuity.

2007-07-05math.GRmath.RT

Simplicity and superrigidity of twin building lattices

Kac-Moody groups over finite fields are finitely generated groups. Most of them can naturally be viewed as irreducible lattices in products of two closed automorphism groups of non-positively curved twinned buildings: those are the most important (but not the only) examples of twin building lattices. We prove that these lattices are simple if and only if the corresponding buildings are (irreducible and) not of affine type (i.e. they are not Bruhat-Tits buildings). In fact, many of them are finitely presented and enjoy property (T). Our arguments explain geometrically why simplicity fails to ho…

2006-09-26math.GR

Superrigidity of actions on finite rank median spaces

Finite rank median spaces are a simultaneous generalisation of finite dimensional ${\rm CAT}(0)$ cube complexes and real trees. If $Γ$ is an irreducible lattice in a product of rank one simple Lie groups, we show that every action of $Γ$ on a complete, finite rank median space has a global fixed point. This is in sharp contrast with the behaviour of actions on infinite rank median spaces. The fixed point property is obtained as corollary to a superrigidity result; the latter holds for irreducible lattices in arbitrary products of compactly generated groups. In previous work, we introduced "Rol…

2019-06-17math.GRmath.GTmath.MG

Cocycle superrigidity from higher rank lattices to $\mathrm{Out}(F_N)$

We prove a rigidity result for cocycles from higher rank lattices to $\mathrm{Out}(F_N)$ and more generally to the outer automorphism group of a torsion-free hyperbolic group. More precisely, let $G$ be either a product of connected higher rank simple algebraic groups over local fields, or a lattice in such a product. Let $G\curvearrowright X$ be an ergodic measure-preserving action on a standard probability space, and let $H$ be a torsion-free hyperbolic group. We prove that every Borel cocycle $G\times X\to\mathrm{Out}(H)$ is cohomologous to a cocycle with values in a finite subgroup of $\ma…

2022-10-11math.GRmath.GT

更多相关论文收起更多论文

Character rigidity for lattices and commensurators

We prove an operator algebraic superrigidity statement for homomorphisms of irreducible lattices, and also their commensurators, in certain higher-rank groups into unitary groups of finite factors. This extends the authors' previous work regarding non-free measure-preserving actions, and also answers a question of Connes for such groups.

2022-03-22math.DSmath.GRmath.OA

At infinity of finite-dimensional CAT(0) spaces

We show that any filtering family of closed convex subsets of a finite-dimensional CAT(0) space $X$ has a non-empty intersection in the visual bordification $ \bar{X} = X \cup \partial X$. Using this fact, several results known for proper CAT(0) spaces may be extended to finite-dimensional spaces, including the existence of canonical fixed points at infinity for parabolic isometries, algebraic and geometric restrictions on amenable group actions, and geometric superrigidity for non-elementary actions of irreducible uniform lattices in products of locally compact groups.

2014-05-15math.GRmath.MG

Splitting fields of real irreducible representations of finite groups

We show that any irreducible representation $ρ$ of a finite group $G$ of exponent $n$, realisable over $\mathbb{R}$, is realisable over the field $E:=\mathbb{Q}(ζ_n)\cap\mathbb{R}$ of real cyclotomic numbers of order $n$, and describe an algorithmic procedure transforming a realisation of $ρ$ over $\mathbb{Q}(ζ_n)$ to one over $E$.

2021-08-07math.GRmath.RT

The minimally displaced set of an irreducible automorphism is locally finite

We prove that the minimally displaced set of a relatively irreducible automorphism of a free splitting, situated in a deformation space, is uniformly locally finite. The minimally displaced set coincides with the train track points for an irreducible automorphism. We develop the theory in a general setting of deformation spaces of free products, having in mind the study of the action of reducible automorphisms of a free group on the simplicial bordification of Outer Space. For instance, a reducible automorphism will have invariant free factors, act on the corresponding stratum of the bordifica…

2020-04-16math.GR

关于分裂域的最小生成集、聚类塔和Galois群的多重传递性

On Minimal generating sets of splitting field, Cluster towers and Multiple transitivity of Galois groups

本文研究了分裂域中最小生成集的性质，揭示了其在Galois群多重传递性下的行为，并将其与根聚类塔概念联系起来。

2026-08-13math.ACmath.COmath.GR

Rigidity of high dimensional graph manifolds

We define the class of high dimensional graph manifolds. These are compact smooth manifolds supporting a decomposition into finitely many pieces, each of which is diffeomorphic to the product of a torus with a finite volume hyperbolic manifold with toric cusps. The various pieces are attached together via affine maps of the boundary tori. We require all the hyperbolic factors in the pieces to have dimension at least 3. Our main goal is to study this class of graph manifolds from the viewpoint of rigidity theory. We show that, in high dimensions, the Borel conjecture holds for our graph manifol…

2013-12-21math.DGmath.GRmath.GT

局部代数的张量分解

Tensor factorization of local algebras

Department of Mathematics and Informatics, Graduate School of Science, Chiba University（千叶大学理学研究科数学与信息学系）

本文证明了分裂域上有限维局部代数的不可约张量分解在同构和顺序意义下是唯一的。

2026-10-01math.GRmath.RA