Algebraic & Geometric Topology

The $L^2$–(co)homology of groups with hierarchies

Boris Okun and Kevin Schreve

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We study group actions on manifolds that admit hierarchies, which generalizes the idea of Haken n–manifolds introduced by Foozwell and Rubinstein. We show that these manifolds satisfy the Singer conjecture in dimensions n 4. Our main application is to Coxeter groups whose Davis complexes are manifolds; we show that the natural action of these groups on the Davis complex has a hierarchy. Our second result is that the Singer conjecture is equivalent to the cocompact action dimension conjecture, which is a statement about all groups, not just fundamental groups of closed aspherical manifolds.

Article information

Algebr. Geom. Topol., Volume 16, Number 5 (2016), 2549-2569.

Received: 5 July 2014
Revised: 24 September 2015
Accepted: 6 April 2016
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]
Secondary: 20J05: Homological methods in group theory

Singer conjecture Haken $n$–manifolds aspherical manifolds Coxeter groups action dimension hierarchy


Okun, Boris; Schreve, Kevin. The $L^2$–(co)homology of groups with hierarchies. Algebr. Geom. Topol. 16 (2016), no. 5, 2549--2569. doi:10.2140/agt.2016.16.2549.

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  • F,D Ancel, C,R Guilbault, $\mathscr Z$–compactifications of open manifolds, Topology 38 (1999) 1265–1280
  • M Bestvina, M Kapovich, B Kleiner, Van Kampen's embedding obstruction for discrete groups, Invent. Math. 150 (2002) 219–235
  • R Charney, M Davis, The Euler characteristic of a nonpositively curved, piecewise Euclidean manifold, Pacific J. Math. 171 (1995) 117–137
  • M,W Davis, Groups generated by reflections and aspherical manifolds not covered by Euclidean space, Ann. of Math. 117 (1983) 293–324
  • M,W Davis, The geometry and topology of Coxeter groups, London Mathematical Society Monographs Series 32, Princeton University Press, NJ (2008)
  • M,W Davis, A,L Edmonds, Euler characteristics of generalized Haken manifolds, Algebr. Geom. Topol. 14 (2014) 3701–3716
  • M,W Davis, I,J Leary, Some examples of discrete group actions on aspherical manifolds, from “High–dimensional manifold topology” (F,T Farrell, W Lück, editors), World Sci., River Edge, NJ (2003) 139–150
  • M,W Davis, B Okun, Vanishing theorems and conjectures for the $\ell\sp 2$–homology of right-angled Coxeter groups, Geom. Topol. 5 (2001) 7–74
  • A,L Edmonds, The Euler characteristic of a Haken $4$–manifold, from “Geometry, groups and dynamics” (C,S Aravinda, W,M Goldman, K Gongopadhyay, A Lubotzky, M Mj, A Weaver, editors), Contemp. Math. 639, Amer. Math. Soc., Providence, RI (2015) 217–234
  • B Foozwell, Haken $n$–manifolds, PhD thesis, University of Melbourne (2007) Available at \setbox0\makeatletter\@url {\unhbox0
  • B Foozwell, The universal covering space of a Haken $n$–manifold, preprint (2011)
  • B Foozwell, H Rubinstein, Introduction to the theory of Haken $n$–manifolds, from “Topology and geometry in dimension three” (W Li, L Bartolini, J Johnson, F Luo, R Myers, J,H Rubinstein, editors), Contemp. Math. 560, Amer. Math. Soc. (2011) 71–84
  • A Grothendieck, Sur quelques points d'algèbre homologique, Tôhoku Math. J. 9 (1957) 119–221
  • C,R Guilbault, Products of open manifolds with $\mathbb R$, Fund. Math. 197 (2007) 197–214
  • S Illman, Existence and uniqueness of equivariant triangulations of smooth proper $G$–manifolds with some applications to equivariant Whitehead torsion, J. Reine Angew. Math. 524 (2000) 129–183
  • J Lott, W Lück, $L\sp 2$–topological invariants of $3$–manifolds, Invent. Math. 120 (1995) 15–60
  • W Lück, $L\sp 2$–invariants: theory and applications to geometry and K-theory, Ergeb. Math. Grenzgeb. 44, Springer (2002)
  • G Mess, Examples of Poincaré duality groups, Proc. Amer. Math. Soc. 110 (1990) 1145–1146
  • J,J Millson, On the first Betti number of a constant negatively curved manifold, Ann. of Math. 104 (1976) 235–247
  • R,S Palais, On the existence of slices for actions of non-compact Lie groups, Ann. of Math. 73 (1961) 295–323
  • G Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint (2002)
  • G Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, preprint (2003)
  • G Perelman, Ricci flow with surgery on three-manifolds, preprint (2003)
  • T,A Schroeder, The $l\sp 2$–homology of even Coxeter groups, Algebr. Geom. Topol. 9 (2009) 1089–1104