A remark on -valued cohomology groups of algebraic group actions

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• 作者：Yongle Jiang ^{yongleji@buffalo.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 22D40 ; 46L10 ; secondary ; 28D05 ; 20J06
• 刊名：Journal of Functional Analysis
• 出版年：2016
• 出版时间：1 August 2016
• 年：2016
• 卷：271
• 期：3
• 页码：577-592
• 全文大小：383 K

文摘

We prove that for a weakly mixing algebraic action class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616000847&_mathId=si175.gif&_user=111111111&_pii=S0022123616000847&_rdoc=1&_issn=00221236&md5=95138e0c43b44a081e7c16cb48c1abd9" title="Click to view the MathML source">σ:G↷(X,ν)class="mathContainer hidden">class="mathCode">$\sigma :G\curvearrowright (X,\nu )$, the n  th cohomology group class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616000847&_mathId=si3.gif&_user=111111111&_pii=S0022123616000847&_rdoc=1&_issn=00221236&md5=a8010883e68341db9ba3fa165684c2d6" title="Click to view the MathML source">Hⁿ(G↷X;T)class="mathContainer hidden">class="mathCode">$H^n(G\curvearrowright X;\mathbb{T})$, after quotienting out the natural subgroup class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616000847&_mathId=si38.gif&_user=111111111&_pii=S0022123616000847&_rdoc=1&_issn=00221236&md5=b2ae0b11a695cdcbd6cd8c9206643cd7" title="Click to view the MathML source">Hⁿ(G,T)class="mathContainer hidden">class="mathCode">$H^n(G,\mathbb{T})$, contains class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616000847&_mathId=si5.gif&_user=111111111&_pii=S0022123616000847&_rdoc=1&_issn=00221236&md5=fae531a1bb52cda4d3dbfa5055fe9e66">class="imgLazyJSB inlineImage" height="21" width="71" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022123616000847-si5.gif">class="mathContainer hidden">class="mathCode">$H^n(G,\hat{X})$ as a natural subgroup for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616000847&_mathId=si6.gif&_user=111111111&_pii=S0022123616000847&_rdoc=1&_issn=00221236&md5=ce3ddfbaf69d1a6b0e4ce26c96a17f41" title="Click to view the MathML source">n=1class="mathContainer hidden">class="mathCode">$n=1$. If we further assume the diagonal actions class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616000847&_mathId=si7.gif&_user=111111111&_pii=S0022123616000847&_rdoc=1&_issn=00221236&md5=067a1481b85f7a3a399676722e06373d" title="Click to view the MathML source">σ²class="mathContainer hidden">class="mathCode">${\sigma }^2$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616000847&_mathId=si8.gif&_user=111111111&_pii=S0022123616000847&_rdoc=1&_issn=00221236&md5=5fda4fe25accaf252bb2f6dd0343d98a" title="Click to view the MathML source">σ⁴class="mathContainer hidden">class="mathCode">${\sigma }^4$ are class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616000847&_mathId=si1.gif&_user=111111111&_pii=S0022123616000847&_rdoc=1&_issn=00221236&md5=242d00a46dbc617d7ca8fb4758e3f484" title="Click to view the MathML source">Tclass="mathContainer hidden">class="mathCode">$\mathbb{T}$-cocycle superrigid and class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616000847&_mathId=si36.gif&_user=111111111&_pii=S0022123616000847&_rdoc=1&_issn=00221236&md5=21dea31f0a1f2924b3a1e5e6a1011627">class="imgLazyJSB inlineImage" height="21" width="70" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022123616000847-si36.gif">class="mathContainer hidden">class="mathCode">$H^2(G,\hat{X})$ is torsion free as an abelian group, then the above also holds true for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616000847&_mathId=si10.gif&_user=111111111&_pii=S0022123616000847&_rdoc=1&_issn=00221236&md5=6ce1fd14144e18f33731a50df77c51bf" title="Click to view the MathML source">n=2class="mathContainer hidden">class="mathCode">$n=2$. Applying it for principal algebraic actions when class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616000847&_mathId=si6.gif&_user=111111111&_pii=S0022123616000847&_rdoc=1&_issn=00221236&md5=ce3ddfbaf69d1a6b0e4ce26c96a17f41" title="Click to view the MathML source">n=1class="mathContainer hidden">class="mathCode">$n=1$, we show that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616000847&_mathId=si11.gif&_user=111111111&_pii=S0022123616000847&_rdoc=1&_issn=00221236&md5=9997f70dd6b77615e386fefdad9beb3a" title="Click to view the MathML source">H²(G,ZG)class="mathContainer hidden">class="mathCode">$H^2(G,\mathbb{Z}G)$ is torsion free as an abelian group when G   has property (T) as a direct corollary of Sorin Popa's cocycle superrigidity theorem; we also use it (when class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616000847&_mathId=si10.gif&_user=111111111&_pii=S0022123616000847&_rdoc=1&_issn=00221236&md5=6ce1fd14144e18f33731a50df77c51bf" title="Click to view the MathML source">n=2class="mathContainer hidden">class="mathCode">$n=2$) to answer, negatively, a question of Sorin Popa on the 2nd cohomology group of Bernoulli shift actions of property (T) groups.