Novikov Conjectures for Arithmetic Groups with Large Actions at Infinity

Abstract

We construct a new compactification of a noncompact rank one globally symmetric space. The result is a nonmetrizable space which also compactifies the Borel-Serre enlargement X of X, contractible only in the appropriateČech sense, and with the action of any arithmetic subgroup of the isometry group of X on X not being small at infinity. Nevertheless, we show that such a compactification can be used in the approach to Novikov conjectures developed recently by G. Carlsson and E. K. Pedersen. In particular, we study the nontrivial instance of the phenomenon of bounded saturation in the boundary of X and deduce that integral assembly maps split in the case of a torsion-free arithmetic subgroup of a semi-simple algebraic Q-group of real rank one or, in fact, the fundamental group of any pinched hyperbolic manifold. Using a similar construction we also split assembly maps for neat subgroups of Hilbert modular groups.

Cech homology and the Novikov Conjectures for K- and L-Theory

“…Boris Goldfarb in his Cornell thesis [10] has veri¢ed these conditions for various groups. Speci¢cally he treats groups À such that À is the (torsion-free) fundamental group of a complete non-compact ¢nite-volume Riemannian manifold with pinched negative sectional curvatures: Àb 2 KM Àa 2`0 .…”

Large scale geometry, compactifications and the integral Novikov conjectures for arithmetic groups

“…The further condition that the universal covering space EΓ of a finite BΓ admits a contractible Γ-compactification with small Γ-action at infinity is not easy to satisfy. See [Go2] for some discussions about this issue.…”

“…In that version, only a special case of Theorem 1.5 was proved and the proof was modeled after [Go2] by using a criterion more relaxed than Theorem 3.2. Since this original approach is probably applicable to prove the integral Novikov conjecture in both K-and L-theories for the mapping class groups (or rather its torsion-free subgroups), we keep an outline of this approach in the current version.…”

“…When S does not contain any non-archimedean place, Γ is an arithmetic subgroup and X S = X. If the rank of X is equal to 1, this problem was solved in [Go2]. It turns out that in general it is difficult to construct a compactification of X BS Q with a small Γ-action at infinity.…”

“…(a) co-compact lattices in connected Lie groups by Carlsson [Ca] (see also [FJ1]), (b) arithmetic subgroups of semisimple linear algebraic groups G defined over Q of R-rank equal to 1 by Goldfarb [Go2], (c) arithmetic subgroups of G = SL(3), and more general arithmetic subgroups of G when G is semisimple and the Q-rank of G is equal to the R-rank of G by Goldfarb [Go3] [ Go1]. (See Remark 6.9 below for comments on the proofs in these papers.…”