A Characterization of Higher Rank Symmetric Spaces Via Bounded Cohomology

By M. Bestvina, K. Fujiwara. 2007

120 citations recorded by the source corpus.

Abstract.Let M be complete nonpositively curved Riemannian manifold of finite volume whose fundamental group Γ does not contain a finite index subgroup which is a product of infinite groups. We show that the universal cover $$\tilde{M}$$ is a higher rank symmetric space iff $$H^{2}_{b} (M; {\mathbb{R}}) \rightarrow H^{2}(M;{\mathbb{R}})$$ is injective (and otherwise the kernel is infinite dimensional). This is the converse of a theorem of Burger–Monod. The proof uses the celebrated Rank Rigidity Theorem, as well as a new construction of quasi-homomorphisms on groups that act on CAT(0) spaces and contain rank 1 elements.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c417cad01a8c0fb615598bcfcaec7967f9fe6261

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In Applied Mathematics

In Geometry and Topology

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Citations in the corpus

Divergence and quasimorphisms of right-angled Artin groups (2010, background): “rphisms, modulo the subspace of true homomorphisms, is denoted QHg(G) and is related to the bounded DIVERGENCE AND QUASIMORPHISMS OF RIGHT-ANGLED ARTIN GROUPS 3 cohomology of G. Bestvina and Fujiwara [BF2] have shown that for group actions on a CAT(0) space, satisfying a weak discontinuity property,”
Asymptotics of Weil–Petersson Geodesics I: Ending Laminations, Recurrence, and Flows (2008, background): “Remark: M. Bestvina and K. Fujiwara have observed indepenently the applicability of this ruled surface technique to the study of action of Mod(S) on Teich(S) as the isometry group of a CAT(0)-space (cf. [ BeFu ]).”
METRIC PROPERTIES OF OUTER SPACE (2008, background): “Is there any analogue of the bounded proje cti n Lemma? (see [3].”
Rank-One Isometries of Buildings and Quasi-Morphisms of Kac–Moody Groups (2008, methodology): “Proof. Follows from Lemma 2.2, Corollary 4.3 and Proposition 4.5, and the fact that for proper CAT(0) spaces, a hyperbolic element is contracting if and only if it is rank one, see [ BeFu2 , Th. 5.4].”
Statistically convex-cocompact actions of groups with contracting elements (2016, background): “[6], [12]; ● certain infinite order elements in GSC, cf.”
Contracting boundaries of CAT(0) spaces (2013, methodology): “In [22], Bestvina and Fujiwara develop many properties of contracting geodesics and, in particular, prove that (1) implies (3).”
Constructing group actions on quasi-trees and applications to mapping class groups (2010, methodology): “e essential feature of this example is that the axis ℓis B-contracting for some B≥ 0. This means that the nearest point projection to ℓof any metric ball disjoint from ℓhas diameter bounded by B. See [BF09]. More generally, one can deﬁne the notion of B-contracting for any subset of a metric space u”