Parameter rigid actions of simply connected nilpotent Lie groups

Maruhashi, Hirokazu

doi:10.1017/etds.2012.112

Cited by 3 publications

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“…Several lemmas before Lemma 37 prepare an 'integration' map µ, which will be used in the proof of Lemma 37. Sublemma 1 inside Lemma 37 is similar to Lemma 43 in the next section, and the same kind of argument already appeared in Maruhashi [16], where the vanishing of H 1 was proved under the assumption of parameter rigidity together with the vanishing of H 0 for actions of nilpotent Lie groups.…”

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Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups, II

Maruhashi

2020

Ergod. Th. Dynam. Sys.

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Let $M\stackrel {\rho _0}{\curvearrowleft }S$ be a $C^\infty $ locally free action of a connected simply connected solvable Lie group S on a closed manifold M. Roughly speaking, $\rho _0$ is parameter rigid if any $C^\infty $ locally free action of S on M having the same orbits as $\rho _0$ is $C^\infty $ conjugate to $\rho _0$ . In this paper we prove two types of result on parameter rigidity. First let G be a connected semisimple Lie group with finite center of real rank at least $2$ without compact factors nor simple factors locally isomorphic to $\mathop {\mathrm {SO}}\nolimits _0(n,1)(n\,{\geq}\, 2)$ or $\mathop {\mathrm {SU}}\nolimits (n,1)(n\geq 2)$ , and let $\Gamma $ be an irreducible cocompact lattice in G. Let $G=KAN$ be an Iwasawa decomposition. We prove that the action $\Gamma \backslash G\curvearrowleft AN$ by right multiplication is parameter rigid. One of the three main ingredients of the proof is the rigidity theorems of Pansu, and Kleiner and Leeb on the quasi-isometries of Riemannian symmetric spaces of non-compact type. Secondly we show that if $M\stackrel {\rho _0}{\curvearrowleft }S$ is parameter rigid, then the zeroth and first cohomology of the orbit foliation of $\rho _0$ with certain coefficients must vanish. This is a partial converse to the results in the author’s [Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups. Geom. Topol. 21(1) (2017), 157–191], where we saw sufficient conditions for parameter rigidity in terms of vanishing of the first cohomology with various coefficients.

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“…THEOREM 21. (Maruhashi [16]) Let N be a connected simply connected nilpotent Lie group, and let M ρ 0 N be a C ∞ locally free action. Then the following are equivalent.…”

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Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups, II

Maruhashi

2020

Ergod. Th. Dynam. Sys.

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“…We will always use Fraktur for the corresponding Lie algebras. The author of this article proved in [4] and [5] the following: Theorem 1. ρ 0 is parameter rigid if and only if H 1 (F ) = H 1 (n).…”

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Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups

Maruhashi

2017

Geom. Topol.

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We give a sufficient condition for parameter rigidity of actions of solvable Lie groups, by vanishing of (uncountably many) first cohomologies of the orbit foliations. In some cases, we can prove that vanishing of finitely many cohomologies is sufficient. For this purpose we use a rigidity property of quasiisometry.As an application we prove some actions of 2-step solvable Lie groups on mapping tori are parameter rigid. Special cases of these actions are considered in a paper of Matsumoto and Mitsumatsu [6].We also remark on the relation between transitive locally free actions of solvable Lie groups and lattices in solvable Lie groups, and apply results in rigidity theory of lattices in solvable Lie groups to construct transitive locally free actions with some properties.

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Deformation of Locally Free Actions and Leafwise Cohomology

Asaoka

2014

Advanced Courses in Mathematics - CRM Barcelona

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Parameter rigid actions of simply connected nilpotent Lie groups

Cited by 3 publications

References 8 publications

Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups, II

Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups, II

Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups

Deformation of Locally Free Actions and Leafwise Cohomology

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