1,384 research outputs found
Splitting formulas for certain Waldhausen Nil-groups
For a group G that splits as an amalgamation of A and B over a common
subgroup C, there is an associated Waldhausen Nil-group, measuring the
"failure" of Mayer-Vietoris for algebraic K-theory. Assume that (1) the
amalgamation is acylindrical, and (2) the groups A,B,G satisfy the
Farrell-Jones isomorphism conjecture. Then we show that the Waldhausen
Nil-group splits as a direct sum of Nil-groups associated to certain
(explicitly describable) infinite virtually cyclic subgroups of G. We note that
a special case of an acylindrical amalgamation includes any amalgamation over a
finite group C.Comment: 12 page
Large scale detection of half-flats in CAT(0)-spaces
For a k-flat F inside a locally compact CAT(0)-space X, we identify various
conditions that ensure that F bounds a (k+1)-dimensional half flat in X. Our
conditions are formulated in terms of the ultralimit of X. As applications, we
obtain (1) constraints on the behavior of quasi-isometries between tocally
compact CAT(0)-spaces, (2) constraints on the possible non-positively curved
Riemannian metrics supported by certain manifolds, and (3) a correspondence
between metric splittings of a complete, simply connected, non-positively
curved Riemannian manifold and the metric splittings of its asymptotic cones.
Furthermore, combining our results with the Ballmann, Burns-Spatzier rigidity
theorem and the classical Mostow rigidity theorem, we also obtain (4) a new
proof of Gromov's rigidity theorem for higher rank locally symmetric spaces.Comment: 21 pages. This article is a substantially improved version of our
earlier preprint arXiv:0801.3636. It features more general results, with
shorter, cleaner proofs. Applications remain the sam
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