Vanishing theorems and conjectures for the ℓ2–homology of right-angled Coxeter groups

Davis, Michael W¹ ; Okun, Boris²

¹ Department of Mathematics, The Ohio State University, Columbus, Ohio 43210, USA
² Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240, USA

Geometry & topology, Tome 5 (2001) no. 1, pp. 7-74.

Résumé

Associated to any finite flag complex $L$ there is a right-angled Coxeter group $W_L$ and a cubical complex ${\Sigma }_L$ on which $W_L$ acts properly and cocompactly. Its two most salient features are that (1) the link of each vertex of ${\Sigma }_L$ is $L$ and (2) ${\Sigma }_L$ is contractible. It follows that if $L$ is a triangulation of ${\mathbb{S}}^{n-1}$, then ${\Sigma }_L$ is a contractible $n$–manifold. We describe a program for proving the Singer Conjecture (on the vanishing of the reduced ${\ell }^2$–homology except in the middle dimension) in the case of ${\Sigma }_L$ where $L$ is a triangulation of ${\mathbb{S}}^{n-1}$. The program succeeds when $n\le 4$. This implies the Charney–Davis Conjecture on flag triangulations of ${\mathbb{S}}^3$. It also implies the following special case of the Hopf–Chern Conjecture: every closed 4–manifold with a nonpositively curved, piecewise Euclidean, cubical structure has nonnegative Euler characteristic. Our methods suggest the following generalization of the Singer Conjecture.

Conjecture: If a discrete group $G$ acts properly on a contractible $n$–manifold, then its ${\ell }^2$–Betti numbers $b_i^{\left(2\right)}\left(G\right)$ vanish for $i>n∕2$.

Détail

DOI : 10.2140/gt.2001.5.7

Keywords: Coxeter group, aspherical manifold, nonpositive curvature, $\ell^2$–homology, $\ell^2$–Betti numbers

Bibliographie

[1] T Akita, Euler characteristics of Coxeter groups, PL–triangulations of closed manifolds, and cohomology of subgroups of Artin groups, J. London Math. Soc. $(2)$ 61 (2000) 721

[2] E M Andreev, Convex polyhedra in Lobačevskiĭspaces, Mat. Sb. $($N.S.$)$ 81 (123) (1970) 445

[3] E M Andreev, Convex polyhedra of finite volume in Lobačevskiĭspace, Mat. Sb. $($N.S.$)$ 83 (125) (1970) 256

[4] M F Atiyah, Elliptic operators, discrete groups and von Neumann algebras, from: "Colloque “Analyse et Topologie” en l'Honneur de Henri Cartan (Orsay, 1974)", Astérisque 32–33, Soc. Math. France (1976) 43

[5] M Bestvina, M Kapovich, B Kleiner, Van Kampen's embedding obstruction for discrete groups, Invent. Math. 150 (2002) 219

[6] N Bourbaki, Éléments de mathématique, Masson (1981) 290

[7] G E Bredon, Sheaf theory, Graduate Texts in Mathematics 170, Springer (1997)

[8] M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften 319, Springer (1999)

[9] W Browder, Surgery on simply-connected manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete 65, Springer (1972)

[10] R Charney, M Davis, The Euler characteristic of a nonpositively curved, piecewise Euclidean manifold, Pacific J. Math. 171 (1995) 117

[11] J Cheeger, M Gromov, Bounds on the von Neumann dimension of $L^2$–cohomology and the Gauss–Bonnet theorem for open manifolds, J. Differential Geom. 21 (1985) 1

[12] J Cheeger, M Gromov, $L_2$–cohomology and group cohomology, Topology 25 (1986) 189

[13] J Cheeger, W Müller, R Schrader, On the curvature of piecewise flat spaces, Comm. Math. Phys. 92 (1984) 405

[14] S S Chern, On curvature and characteristic classes of a Riemann manifold, Abh. Math. Sem. Univ. Hamburg 20 (1955) 117

[15] M W Davis, Groups generated by reflections and aspherical manifolds not covered by Euclidean space, Ann. of Math. $(2)$ 117 (1983) 293

[16] M W Davis, Coxeter groups and aspherical manifolds, from: "Algebraic topology, Aarhus 1982 (Aarhus, 1982)", Lecture Notes in Math. 1051, Springer (1984) 197

[17] M W Davis, Nonpositive curvature and reflection groups, from: "Handbook of geometric topology", North-Holland (2002) 373

[18] M Davis, T Januszkiewicz, R Scott, Nonpositive curvature of blow-ups, Selecta Math. $($N.S.$)$ 4 (1998) 491

[19] M W Davis, G Moussong, Notes on nonpositively curved polyhedra, from: "Low dimensional topology (Eger, 1996/Budapest, 1998)", Bolyai Soc. Math. Stud. 8, János Bolyai Math. Soc. (1999) 11

[20] J Dodziuk, de Rham–Hodge theory for $L^2$–cohomology of infinite coverings, Topology 16 (1977) 157

[21] J Dodziuk, $L^2$ harmonic forms on rotationally symmetric Riemannian manifolds, Proc. Amer. Math. Soc. 77 (1979) 395

[22] H Donnelly, F Xavier, On the differential form spectrum of negatively curved Riemannian manifolds, Amer. J. Math. 106 (1984) 169

[23] B Eckmann, Introduction to $l_2$–methods in topology: reduced $l_2$–homology, harmonic chains, $l_2$–Betti numbers, Israel J. Math. 117 (2000) 183

[24] M S Farber, Homological algebra of Novikov–Shubin invariants and Morse inequalities, Geom. Funct. Anal. 6 (1996) 628

[25] M Farber, von Neumann Betti numbers and Novikov type inequalities, Proc. Amer. Math. Soc. 128 (2000) 2819

[26] M Gromov, Hyperbolic groups, from: "Essays in group theory", Math. Sci. Res. Inst. Publ. 8, Springer (1987) 75

[27] M Gromov, Kähler hyperbolicity and $L_2$–Hodge theory, J. Differential Geom. 33 (1991) 263

[28] M Gromov, Asymptotic invariants of infinite groups, from: "Geometric group theory, Vol. 2 (Sussex, 1991)", London Math. Soc. Lecture Note Ser. 182, Cambridge Univ. Press (1993) 1

[29] J L Gross, T W Tucker, Topological graph theory, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley Sons (1987)

[30] A Haefliger, Complexes of groups and orbihedra, from: "Group theory from a geometrical viewpoint (Trieste, 1990)", World Sci. Publ., River Edge, NJ (1991) 504

[31] H Kesten, Full Banach mean values on countable groups, Math. Scand. 7 (1959) 146

[32] J Lott, W Lück, $L_2$–topological invariants of 3–manifolds, Invent. Math. 120 (1995) 15

[33] W Lück, $L_2$–Betti numbers of mapping tori and groups, Topology 33 (1994) 203

[34] J J Millson, On the first Betti number of a constant negatively curved manifold, Ann. of Math. $(2)$ 104 (1976) 235

[35] G Moussong, Hyperbolic Coxeter groups, PhD thesis, Ohio State University (1988)

[36] D Sullivan, Travaux de Thurston sur les groupes quasi-fuchsiens et les variétés hyperboliques de dimension 3 fibrées sur $S^1$, from: "Bourbaki Seminar, Vol. 1979/80", Lecture Notes in Math. 842, Springer (1981) 196

[37] W P Thurston, A norm for the homology of 3–manifolds, Mem. Amer. Math. Soc. 59 (1986) 99

[38] W P Thurston, Orbifolds and Seifert Fiber Spaces, (1990)

[39] C T C Wall, Rational Euler characteristics, Proc. Cambridge Philos. Soc. 57 (1961) 182