Citation
Qian, Nantian (1992) Rigidity Phenomena of Group Actions on a Class of Nilmanifolds and Anosov R^n Actions. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/M7JXZV02. https://resolver.caltech.edu/CaltechTHESIS:07122011075820059
Abstract
An action of a group Г on a manifold M is a homomorphism ρ from Г to Diff(M). ρo is locally rigid if the nearby homomorphism ρ, ρ(γ) = h o ρ0, (γ) 0h^(1) for some h Є Diff(M) and for all, γ Є Г. In other words, ρ0 is isolated from other actions up to a smooth conjugation. In this thesis we studied some standard group actions on a broader class of manifolds, the free, kstep nilmanifolds N(n, k); we obtained that the standard SL(n, Z) action on N(n,2) is locally rigid for n = 3, and n ≥ 5. We recall that N(n,1) = T^n. Hence, our results are the generalization to the local rigidity result for the standard action on torus T^n. We observed also, for the first time, that for discrete subgroups Aut(n, 2) of a Lie group, which is not even reductive, the action on N(n,2) is deformationrigid for n = 3, and n ≥ 5. We also investigated the dynamics of Anosov R^n actions and obtained a number of results parallel to those of Anosov diffeomorphisms and flows. E.g., the strong stable (unstable) manifold for a regular element is dense iff the action is weakly mixing (for a volumepreserving action); an Anosov action with no dense, strong stable (unstable) manifold can always be reduced to suspension of the action mentioned above; there are two compatible measures to the Anosov actions.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  Mathematics 
Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  22 April 1992 
Record Number:  CaltechTHESIS:07122011075820059 
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:07122011075820059 
DOI:  10.7907/M7JXZV02 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  6538 
Collection:  CaltechTHESIS 
Deposited By:  John Wade 
Deposited On:  12 Jul 2011 15:21 
Last Modified:  21 Dec 2019 02:06 
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