TY - JOUR
AU1 - Ioana, Adrian
AU2 - Spaas, Pieter
AU3 - Wiersma, Matthew
AB - We develop a new method, based on non-vanishing of second cohomology groups, for proving the failure of lifting properties for full C∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$^*$$\end{document}-algebras of countable groups with (relative) property (T). We derive that the full C∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$^*$$\end{document}-algebras of the groups Z2×SL2(Z)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {Z}^2\times \text {SL}_2({\mathbb {Z}})$$\end{document} and SLn(Z)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\text {SL}_n({\mathbb {Z}})$$\end{document}, for n≥3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n\ge 3$$\end{document}, do not have the local lifting property (LLP). We also prove that the full C∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$^*$$\end{document}-algebras of a large class of groups Γ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma $$\end{document} with property (T), including those such that H2(Γ,R)≠0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\text {H}^2(\Gamma ,{\mathbb {R}})\not =0$$\end{document} or H2(Γ,ZΓ)≠0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\text {H}^2(\Gamma ,\mathbb {Z}\Gamma )\not =0$$\end{document}, do not have the lifting property (LP). More generally, we show that the same holds if Γ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma $$\end{document} admits a probability measure preserving action with non-vanishing second R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {R}}$$\end{document}-valued cohomology. Finally, we prove that the full C∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$^*$$\end{document}-algebra of any non-finitely presented property (T) group fails the LP.
TI - Cohomological obstructions to lifting properties for full C∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepa ...
JF - Geometric and Functional Analysis
DO - 10.1007/s00039-020-00550-4
DA - 2020-10-26
UR - https://www.deepdyve.com/lp/springer-journals/cohomological-obstructions-to-lifting-properties-for-full-c-nrdMSd2NFT
SP - 1402
EP - 1438
VL - 30
IS - 5
DP - DeepDyve
ER -