By T. W. Müller

A few eminent mathematicians have been invited to Bielefeld, Germany in 1999 to give lectures at a convention on topological, combinatorial and mathematics features of (infinite) teams. the current quantity includes survey and learn articles invited from members during this convention. The contributions are geared to experts and aspiring graduate and post-graduate scholars attracted to pursuing extra examine.

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Definition 1. Let G be a group with two BN–pairs (B+ , N ), (B− , N ) with the same subgroup N and such that B+ ∩ N = B− ∩ N . Denote by W the common Weyl group W = N /B+ ∩ N = N /B− ∩ N , and let S be the distinguished set of generators of W. We denote the length function on W with respect to S by l. Then (B+ , B− , N ) is called a twin BN–pair in G if the following two conditions are satisfied. (TBN1) B w B− s B− = B ws B− for ∈ {+, −}, all w ∈ W, and all s ∈ S such that l(ws) < l(w), (TBN2) B+ s ∩ B− = ∅ for all s ∈ S.

Bieri∗ and R. Geoghegan† This is a report on our work during the last few years on extending the BieriNeumann-Strebel-Renz theory of “geometric invariants” of groups to a theory of group actions on non-positively curved (= CAT(0)) spaces. 3, proofs of all our theorems can be found in our papers [BGI ] (controlled connectivity and openness results), [BGII ] (the geometric invariants) and [BGIII ] (S L 2 actions on the hyperbolic plane). An earlier expository paper [BG 98] is also relevant. 1. The geometric invariants Here we recall the “geometric” or “ -” invariants of groups developed during the 1980’s by Bieri, Neumann, Strebel and Renz (abbrev.

The vector space V has a natural base point 0. An endpoint of V is a ray starting at 0. The set, ∂ V , of endpoints is a sphere of dimension (dim V ) −1 with the obvious topology. We choose an inner product (·, ·) for V . Then each e ∈ ∂ V defines a “projection on the e-direction” functional πe ∈ V ∗ by πe (x) = (x, u e ), where u e is the point on the ray e of unit distance from 0. Thus we get, for e ∈ ∂ V , a character χe = πe ◦ α : G → R. The map ∂ V → S(G), e → [χe ] is a homeomorphism. Looking ahead, we say that the action α is CC n−1 over e ∈ ∂ V if and only if χe is CC n−1 over ∞.