Coarse Obstructions to Positive Scalar Curvature in Noncompact Arithmetic Manifolds

Abstract

ABSTRACT. Block and Weinberger show that an arithmetic manifold can be endowed with a positive scalar curvature metric if and only if its Q-rank exceeds 2. We show in this article that these metrics are never in the same coarse class as the natural metric inherited from the base Lie group. Furthering the coarse C * -algebraic methods of Roe, we find a nonzero Dirac obstruction in the Ktheory of a particular operator algebra which encodes information about the quasi-isometry type of the manifold as well as its local geometry.

Reduction theory for mapping class groups and applications to moduli spaces

“…These Riemannian metrics have the quasi-isometry type of rays and one may ask whether there is also a metric that is both uniformly positively curved and quasi-isometric to the symmetric metric. That question was answered negatively for locally symmetric spaces associated to arithmetic groups of Q-rank ≥ 2 by Chang (see [6]). Farb and Weinberger recently anounced a result analogous to [3] for M(S): A finite cover of the moduli space admits a complete, finite volume Riemannian metric of uniformly bounded positive scalar curvature if and only if d(S) ≥ 3 (see [7], 4.2).…”

“…We therefore only sketch those ingredients which rely on general results and in particular refer to [6] for definitions and more details.…”

“…Then CO(x) is contained in a subset of M(S) diffeomeorphic to B ε 0 (τ ) × R (A, B) forms a "coarsely excicive decomposition" of M(S) with intersection A ∩ B = V (see [6] for the definition).…”

“…Then one defines a generalized Roe algebra C * Γ M(S) and a generalized coarse index Ind(D) ∈ K * (C * Γ M(S)) as in [6].…”

Topological nonrigidity of nonuniform lattices

“…If \G/K is noncompact, Borel and Serre [3] construct a well-known compactification M of M whose -cover has boundary homotopy equivalent to a countably infinite wedge of (r − 1)-spheres, where r is the rational rank of G. Recall that, if G is a Q-subgroup of SL n (R) and is commensurable with G Z , then the rational rank rank Q ( ) of G is the dimension of any maximal Q-split torus of G. In fact, certain curvature and rigidity phenomena occur or fail to occur in arithmetic manifolds in accordance with the size of its rational rank. Block and Weinberger [4] prove that M = \G/K admits a metric of positive scalar curvature if and only if rank Q ( ) ≥ 3, although such positively curved metrics never belong to the same coarse class as the natural metric on M inherited from the Lie group structure of G (see Chang [5]). …”