Alexander Gorodnik
Let Γbe a lattice in G=SL(n,R) and X=G/S a homogeneous space of G, where S is a closed subgroup of G which contains a real algebraic subgroup H such that G/H is compact. We establish uniform distribution of orbits of Γin X analogous to the classical equidistribution on torus. To obtain this result, we first prove an ergodic theorem along balls in the connected component of Borel subgroup of G.
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Let $Γ$ be a lattice in ${\rm SL}(n, \mathbb R)$ with $n\geq 3$ and $\mathcal S$ be a closed surface. Then $Γ$ has no distal minimal action on $\mathcal S$.
Partially hyperbolic lattice actions on 2-step nilmanifolds
本文研究高阶秩格在2步幂零流形上包含部分双曲元素的作用，在技术假设下证明其必为仿射映射，从而推广了Brown等人的刚性结果。
Every stationary action of a strongly irreducible lattice or commensurator of such a latiice in a general semisimple group, with at least one higher-rank connected factor, either has finite stabilizers almost surely or finite index stabilizers almost surely.
We study the connection between the Lyapunov exponents and the volume growth of boundary distortion of regions in the phase space of the dynamical system.
Boundary actions by higher-rank lattices: Classification and embedding in low dimensions, local rigidity, smooth factors
本文研究了高秩格群在紧流形上的作用，通过分类相关不变测度推导出新的刚性结果，证明了低维作用共轭于边界作用，并验证了标准边界作用的局部刚性及光滑因子的分类。
In this paper we study Zimmer's conjecture for $C^1$ actions of lattice subgroup of a higher-rank simple Lie group with finite center on compact manifolds. We show that when the rank of an uniform lattice is larger than the dimension of the manifold, then the action factors through a finite group. For lattices in $SL(n, \R)$, the dimensional bound is sharp.
We study smooth factors of the standard actions of lattices in higher rank semisimple Lie groups on flag manifolds. Under a mild condition on existence of a single differentiable sink, we show that these factors are $C^{\infty}$-conjugate to the standard actions on flag manifolds.
The mapping class group of a compact oriented surface of genus greater than one with boundary acts ergodically on connected components of the representation variety corresponding to a connected compact Lie group, for every choice of conjugacy class boundary condition.
In this article we prove global rigidity results for hyperbolic actions of higher-rank lattices. Suppose $Γ$ is a lattice in semisimple Lie group, all of whose factors have rank $2$ or higher. Let $α$ be a smooth $Γ$-action on a compact nilmanifold $M$ that lifts to an action on the universal cover. If the linear data $ρ$ of $α$ contains a hyperbolic element, then there is a continuous semiconjugacy intertwining the actions of $α$ and $ρ$, on a finite-index subgroup of $Γ$. If $α$ is a $C^\infty$ action and contains an Anosov element, then the semiconjugacy is a $C^\infty$ conjugacy. As a coro…
We study the rigidity in the sense of Zimmer for higher rank lattice actions on dendrites and show that: (1) if $Γ$ is a higher rank lattice and $X$ is a nondegenerate dendrite with no infinite order points, then any action of $Γ$ on $X$ cannot be almost free; (2) if $Γ$ is further a finite index subgroup of $SL_n(\mathbb Z)$ with $n\geq 3$, then every action of $Γ$ on $X$ has a nontrivial almost finite subsystem. During the proof, we get a new characterization of the left-orderability of a finitely generated group through its actions on dendrites.
We consider smooth actions of lattices in higher-rank semisimple Lie groups on manifolds. We define two numbers $r(G)$ and $m(G)$ associated with the roots system of the Lie algebra of a Lie group $G$. If the dimension of the manifold is smaller than $r(G)$, then we show the action preserves a Borel probability measure. If the dimension of the manifold is at most $m(G)$, we show there is a quasi-invariant measure on the manifold such that the action is measurable isomorphic to a relatively measure preserving action over a standard boundary action.
We prove that any real-analytic, volume-preserving action of a lattice $Γ$ in a simple Lie group with $\Qrank(Γ)\geq 7$ on a closed 4-manifold of nonzero Euler characteristic factors through a finite group action.
We prove Zimmer's conjecture for co-compact lattices in ${\rm SL}(n, \mathbb C)$: for any co-compact lattice in ${\rm SL}(n, \mathbb C)$, $n \geq 3$, any $Γ$-action on a compact manifold $M$ with dimension: (I) less than $2n-2$ if $n \neq 4$, (II) less than $5$ if $n = 4$, by $C^{1+ε}$ diffeomorphisms factors through a finite action.
We give a topological stability result for the action of the fundamental group of a compact manifold of negative curvature on its boundary at infinity: any nearby action of this group by homeomorphisms of the sphere is semi-conjugate to the standard boundary action. Using similar techniques we prove a global rigidity result for the "slithering actions" of 3-manifold groups that come from skew-Anosov flows. As applications, we construct hyperbolic 3-manifolds that admit arbitrarily many topologically inequivalent Anosov flows, answering a question from Kirby's problem list, and also give a more…
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.
We investigate conformal actions of cocompact lattices in higher-rank simple Lie groups on compact pseudo-Riemannian manifolds. Our main result gives a general bound on the real-rank of the lattice, which was already known for the action of the full Lie group by a result of Zimmer. When the real-rank is maximal, we prove that the manifold is conformally flat. This indicates that a global conclusion similar to that of anterior works of Bader, Nevo and Frances, Zeghib in the case of a Lie group action might be obtained. We also give better estimates for actions of cocompact lattices in exception…
We prove that any ergodic measure-preserving action of an irreducible lattice in a semisimple group, with finite center and each simple factor having rank at least two, either has finite orbits or has finite stabilizers. The same dichotomy holds for many commensurators of such lattices. The above are derived from more general results on groups with the Howe-Moore property and property $(T)$. We prove similar results for commensurators in such groups and for irreducible lattices (and commensurators) in products of at least two such groups, at least one of which is totally disconnected.
We consider actions of cocompact lattices in semisimple Lie groups of the noncompact type on their boundaries $G/Q$, $Q$ a parabolic group, the so-called standard actions. We show that perturbations of the standard action in the homeomorphism group continuously factor onto the original standard action by a semi-conjugacy close to the identity. This generalizes works by Bowden, Mann, Manning and Weisman in the setting of negative curvature or Gromov hyperbolic groups. Finally, we also construct perturbations of the action of lattices on the geodesic boundary which are not $C^0$ semi-conjugate t…
Hyperfiniteness and Borel asymptotic dimension of boundary actions of hyperbolic groups
本文为有限生成双曲群 Gromov 边界作用的轨道等价关系超有限性定理提供新的简短证明，并进一步证明此类作用具有有限 Borel 渐近维数。
Boundary actions of outer automorphism groups of Thompson-like groups
本文研究了索恩-汤普森群的外自同构群的边界作用，并证明这些群是C*-简单。
We prove that any smooth volume-preserving action of a lattice $Γ$ in $\textrm{SL}(n,\mathbb{R})$, $n\ge 3$, on a closed $n$-manifold which possesses one element that admits a dominated splitting should be standard. In other words, the manifold is the $n$-flat torus and the action is smoothly conjugate to an affine action. Note that an Anosov diffeomorphism, or more generally, a partial hyperbolic diffeomorphism admits a dominated splitting. We have a topological global rigidity when $α$ is $C^{1}$. Similar theorems hold for an action of a lattice in $\textrm{Sp}(2n,\mathbb{R})$ with $n\ge 2$ …
Boundary actions of CAT(0) spaces: topological freeness and applications to $C^*$-algebras
本文通过验证 Myrberg 点的自由性，证明 CAT(0) 空间的多类边界作用（视觉、horofunction、Roller）是拓扑自由的强边界作用，推广了已知结果，并应用于构造新的 $C^*$-无自群及精确纯无限单的约化交叉积 $C^*$-代数。
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