Cohomological equation and cocycle rigidity of discrete parabolic actions in some higher-rank Lie groups

Tanis, James; Wang, Zhenqi Jenny

doi:10.1007/s11854-020-0136-1

Cited by 3 publications

(19 citation statements)

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“…The estimates of the solution in above theorem are not tame, i.e., there is no finite loss of regularity (with respect to Sobolev norms) between the coboundary and the solution. Similar results were proven for the (discrete) classical horocycle map, see [9], [38] and [39]. The next result shows that when G = SL(n, R), n ≥ 3, they are indeed the best possible up to a finite loss of regularity.…”

Section: 3supporting

confidence: 76%

“…Theorem 2.2 proves that in the case of SL(n, R), for n ≥ 3, these Sobolev estimates are, in fact, generally not tame, and our Sobolev estimates in Theorem 2.1 are sharp up to finite loss of regularity. Theorem 2.2 gives the same lower bound for SL(2, R), which is part of the proof of analogous sharp (up to finite loss of regularity), non-tame estimates for horocycle maps, to appear in the forthcoming paper [39].…”

mentioning

confidence: 67%

“…For example, the groups SO 0 (2, 2) and SO o (3,3) are locally isomorphic to SL(2, R) × SL(2, R) and SL(4, R) respectively. Cohomology properties for lower rank cases will appear in a forthcoming paper, see [39].…”

Section: Resultsmentioning

confidence: 99%

“…Notice that Theorem 2.1 holds for the regular representation H = L 2 0 (G/Γ), consisting of square integrable functions on G/Γ orthogonal to constants, where Γ is an arbitrary lattice. Our proof for this general lattice uses a analogous estimate for unitary representations of SL(2, R), which will appear in [39]. However, when Γ is cocompact, the estimate for SL(2, R) representations is not needed, because…”

Section: 3mentioning

confidence: 99%

“…For example, the space of obstructions to a smooth solution of the cohomological equation of horocycle maps has infinite countable dimension in each irreducible component of P SL(2, R), see [38], as opposed to being at most two dimensional in each component for the horocycle flow, see [8]. Moreover, Sobolev estimates for solutions of the cohomological equation of horocycle maps are not tame in [38], [39] and [9], as opposed to the tame estimates obtained for the horocycle flow in [8] and parabolic flow in [42]. Because tame estimates for solutions of the cohomological equation lays the groundwork for proving smooth action rigidity, see [5] and [6], not having them complicates this effort.…”

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confidence: 99%

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Cohomological equation and cocycle rigidity of discrete parabolic actions in some higher rank Lie groups

Tanis¹,

Wang²

2018

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Let G denote a higher rank R-split simple Lie group of the following type: SL(n, R), SOo(m, m), E 6( 6) , E 7( 7) and E 8(8) , where m ≥ 4 and n ≥ 3. We study the cohomological equation for discrete parabolic actions on G via representation theory. Specifically, we characterize the obstructions to solving the cohomological equation and construct smooth solutions with Sobolev estimates. We prove that global estimates of the solution are generally not tame, and our non-tame estimates in the case G = SL(n, R) are sharp up to finite loss of regularity. Moreover, we prove that for general G the estimates are tame in all but one direction, and as an application, we obtain tame estimates for the common solution of the cocycle equations. We also give a sufficient condition for which the first cohomology with coefficients in smooth vector fields is trivial. In the case that G = SL(n, R), we show this condition is also necessary. A new method is developed to prove tame directions involving computations within maximal unipotent subgroups of the unitary duals of SL(2, R) ⋉ R 2 and SL(2, R) ⋉ R 4 . A new technique is also developed to prove non-tameness for solutions of the cohomological equation.

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Section: 3supporting

confidence: 76%

mentioning

confidence: 67%

Section: Resultsmentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

mentioning

confidence: 99%

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Cohomological equation and cocycle rigidity of discrete parabolic actions in some higher rank Lie groups

Tanis¹,

Wang²

2018

Preprint

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Transversal local rigidity of discrete abelian actions on Heisenberg nilmanifolds

Damjanović¹,

Tanis

2021

Ergod. Th. Dynam. Sys.

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Cohomological equation and cocycle rigidity of discrete parabolic actions

Tanis¹,

Wang²

2019

Discrete &Amp; Continuous Dynamical Systems - A

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We study the cohomological equation for discrete horocycle maps on SL(2, R) and SL(2, R) × SL(2, R) via representation theory. Specifically, we prove Hilbert Sobolev non-tame estimates for solutions of the cohomological equation of horocycle maps in representations of SL(2, R). Our estimates improve on previous results and are sharp up to a fixed, finite loss of regularity. Moreover, they are tame on a codimension one subspace of sl(2, R), and we prove tame cocycle rigidity for some two-parameter discrete actions, improving on a previous result. Our estimates on the cohomological equation of horocycle maps overcome difficulties in previous papers by working in a more suitable model for SL(2, R) in which all cases of irreducible, unitary representations of SL(2, R) can be studied simultaneously.Finally, our results combine with those of a very recent paper by the authors to give cohomology results for discrete parabolic actions in regular representations of some general classes of simple Lie groups, providing a fundamental step toward proving differential local rigidity of parabolic actions in this general setting.

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Cohomological equation and cocycle rigidity of discrete parabolic actions in some higher-rank Lie groups

Cited by 3 publications

References 36 publications

Cohomological equation and cocycle rigidity of discrete parabolic actions in some higher rank Lie groups

Cohomological equation and cocycle rigidity of discrete parabolic actions in some higher rank Lie groups

Transversal local rigidity of discrete abelian actions on Heisenberg nilmanifolds

Cohomological equation and cocycle rigidity of discrete parabolic actions

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