This is a report on our work during the last few years on extending the Bieri-Neumann-Strebel-Renz theory of “geometric invariants” of groups to a theory of group actions on non-positively curved (= CAT(0)) spaces. With the exception of Theorem 8, which is proved here, and the related material in §5.3, proofs of all our theorems can be found in our papers [BGI] (controlled connectivity and openness results), [BGII] (the geometric invariants) and [BGIII] (SL2 actions on the hyperbolic plane). An earlier expository paper [BG 98] is also relevant.

The geometric invariants

we recall the “geometric” or “Σ-” invariants of groups developed during the 1980's by Bieri, Neumann, Strebel and Renz (abbrev. BNSR); see [BNS 87], [BR 88], [Re 88]. We set things out in a way which leads directly to generalizations which were not anticipated in the original literature. Let G be a group of type Fn, n ≥ 1. Let X be a contractible G-CW complex which is either (a) free with cocompact n-skeleton, or (b) properly discontinuous and cocompact. Case (a) exists by the definition of Fn; Case (b) is often useful but can only exist when G has finite virtual cohomological dimension.

Controlled connectivity

Let χ : G → ℝ be a non-zero character, i.e., a homomorphism to the additive group of real numbers. Reinterpret ℝ as the group of translations, Transl, of the Euclidean line, and thus reinterpret χ as an action of G on by translations.