Group homology and connes periodicity operator

Emmanouil, Ioannis ; Passi, Inder Bir S. (2006) Group homology and connes periodicity operator Journal of Pure and Applied Algebra, 205 (2). pp. 375-392. ISSN 0022-4049

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Official URL: http://www.sciencedirect.com/science/article/pii/S...

Related URL: http://dx.doi.org/10.1016/j.jpaa.2005.07.008

Abstract

Given a commutative ring k, a group G and an element gε G of infinite order with centralizer C(g), we study the inverse system ....→ H2n(C(g)/<g>,k)→H2n-2(C(g)/<g>,k) →...arising from Burghelea's computation [D. Burghelea, The cyclic homology of group rings, Comment. Math. Helv. 60 (1985) 354-365] of the cyclic homology of the group algebra kG and Connes' periodicity operator S:HC2n(kG)→ HC2n-2(kG). A vanishing theorem for the limit of this inverse system is proved for groups in the class introduced in Emmanouil and Passi [A contribution to Bass' conjecture, J. Group Theory 7 (2004) 409-420], thereby contributing to a conjecture by Burghelea [The cyclic homology of group rings, Comment. Math. Helv. 60 (1985) 354-365]. The homological condition defining the class is closely examined; in particular, it is shown that this class properly contains the class studied in Emmanouil [On a class of groups satisfying Bass' conjecture, Invent. Math. 132 (1998) 307-330].

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