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

Maruhashi, Hirokazu

doi:10.2140/gt.2017.21.157

Cited by 1 publication

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“…(1) Vanishing of cohomology ⇒ parameter rigidity. This is the sufficient condition for parameter rigidity proved in [17]. In the current article this is Theorem 4.…”

supporting

confidence: 55%

“…This section is a summary of the results we need later, proved in Maruhashi [17]. See [17] for the details.…”

Section: Preliminariesmentioning

confidence: 92%

“…This section is a summary of the results we need later, proved in Maruhashi [17]. See [17] for the details. In this paper Lie algebras are denoted by the corresponding lowercase Fraktur of the corresponding Lie groups.…”

Section: Preliminariesmentioning

confidence: 92%

“…We give a proof of this theorem in §4 and §5 after recalling the results in Maruhashi [17] in §3. The proof is a combination of the following three steps.…”

mentioning

confidence: 98%

“…The author does not know whether such S and Ŵ exist. But if we have such a pair, Maruhashi [17,Proposition 4] says that the action Ŵ\S S defined by right multiplication is parameter rigid, because in this case parameter rigidity is equivalent to the rigidity As a more concrete example, we consider the two-dimensional solvable Lie algebra ga = RX ⊕ RY defined by [X, Y ] = Y . Then the one-dimensional representation ga ad RY satisfies the condition of Theorem 32, but the trivial representation s ga/RY does not satisfy the condition.…”

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confidence: 99%

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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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“…(1) Vanishing of cohomology ⇒ parameter rigidity. This is the sufficient condition for parameter rigidity proved in [17]. In the current article this is Theorem 4.…”

supporting

confidence: 55%

“…This section is a summary of the results we need later, proved in Maruhashi [17]. See [17] for the details.…”

Section: Preliminariesmentioning

confidence: 92%

Section: Preliminariesmentioning

confidence: 92%

“…We give a proof of this theorem in §4 and §5 after recalling the results in Maruhashi [17] in §3. The proof is a combination of the following three steps.…”

mentioning

confidence: 98%

mentioning

confidence: 99%

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