## Algebraic & Geometric Topology

### The fattened Davis complex and weighted $L^2$–(co)homology of Coxeter groups

Wiktor Mogilski

#### Abstract

This article consists of two parts. First, we propose a program to compute the weighted $L2$–(co)homology of the Davis complex by considering a thickened version of this complex. The program proves especially successful provided that the weighted $L2$–(co)homology of certain infinite special subgroups of the corresponding Coxeter group vanishes in low dimensions. We then use our complex to perform computations for many examples of Coxeter groups. Second, we prove the weighted Singer conjecture for Coxeter groups in dimension three under the assumption that the nerve of the Coxeter group is not dual to a hyperbolic simplex, and in dimension four under the assumption that the nerve is a flag complex. We then prove a general version of the conjecture in dimension four where the nerve of the Coxeter group is assumed to be a flag triangulation of a $3$–manifold.

#### Article information

Source
Algebr. Geom. Topol., Volume 16, Number 4 (2016), 2067-2105.

Dates
Revised: 27 October 2015
Accepted: 12 November 2015
First available in Project Euclid: 28 November 2017

https://projecteuclid.org/euclid.agt/1511895908

Digital Object Identifier
doi:10.2140/agt.2016.16.2067

Mathematical Reviews number (MathSciNet)
MR3546459

Zentralblatt MATH identifier
06627569

#### Citation

Mogilski, Wiktor. The fattened Davis complex and weighted $L^2$–(co)homology of Coxeter groups. Algebr. Geom. Topol. 16 (2016), no. 4, 2067--2105. doi:10.2140/agt.2016.16.2067. https://projecteuclid.org/euclid.agt/1511895908

#### References

• E,M Andreev, On convex polyhedra of finite volume in Lobachevskii space, Mat. Sb. 12 (1971) 256–260 In Russian; translated in Math. USSR-Sb. 12 (1970) 255–259
• M Bestvina, The virtual cohomological dimension of Coxeter groups, from: “Geometric group theory, 1”, (G,A Niblo, M,A Roller, editors), London Math. Soc. Lecture Note Ser. 181, Cambridge Univ. Press (1993) 19–23
• K,S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer, New York (1982)
• J Cheeger, M Gromov, $L\sb 2$–cohomology and group cohomology, Topology 25 (1986) 189–215
• 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 Univ. Press (2008)
• M,W Davis, J Dymara, T Januszkiewicz, J Meier, B Okun, Compactly supported cohomology of buildings, Comment. Math. Helv. 85 (2010) 551–582
• M,W Davis, J Dymara, T Januszkiewicz, B Okun, Weighted $L^2$–cohomology of Coxeter groups, Geom. Topol. 11 (2007) 47–138
• M,W Davis, B Okun, Vanishing theorems and conjectures for the $\ell^2$–homology of right-angled Coxeter groups, Geom. Topol. 5 (2001) 7–74
• M,W Davis, B Okun, Cohomology computations for Artin groups, Bestvina–Brady groups, and graph products, Groups Geom. Dyn. 6 (2012) 485–531
• J Dodziuk, $L\sp{2}$ harmonic forms on rotationally symmetric Riemannian manifolds, Proc. Amer. Math. Soc. 77 (1979) 395–400
• J Dymara, $L\sp 2$–cohomology of buildings with fundamental class, Proc. Amer. Math. Soc. 132 (2004) 1839–1843
• J Dymara, Thin buildings, Geom. Topol. 10 (2006) 667–694
• J Dymara, T Januszkiewicz, Cohomology of buildings and their automorphism groups, Invent. Math. 150 (2002) 579–627
• D,B,A Epstein, A Marden, Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces, from: “Analytical and geometric aspects of hyperbolic space”, (D,B,A Epstein, editor), London Math. Soc. Lecture Note Ser. 111, Cambridge Univ. Press (1987) 113–253
• J,E Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics 29, Cambridge Univ. Press (1990)
• 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, Berlin (2002)
• B Okun, K Schreve, The $L^2$–(co)homology of groups with hierarchies, preprint (2014)
• G Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, preprint (2003)
• T,A Schroeder, The $l^2$–homology of even Coxeter groups, Algebr. Geom. Topol. 9 (2009) 1089–1104
• T,A Schroeder, On the three-dimensional Singer conjecture for Coxeter groups, preprint (2009)
• T,A Schroeder, $\ell\sp 2$–homology and planar graphs, Colloq. Math. 131 (2013) 129–139