Fixed points and amenability in non-positive curvature

Caprace, Pierre-Emmanuel;Monod, Nicolas
(2013) Mathematische Annalen — Vol. 356, n° 4, p. 1303-1337 (2013)

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Abstract
Consider a proper cocompact CAT(0) space X. We give a complete algebraic characterisation of amenable groups of isometries of X. For amenable discrete subgroups, an even narrower description is derived, implying Q-linearity in the torsion-free case. We establish Levi decompositions for stabilisers of points at infinity of X, generalising the case of linear algebraic groups to Is(X). A geometric counterpart of this sheds light on the refined bordification of X (à la Karpelevich) and leads to a converse to the Adams-Ballmann theorem. It is further deduced that unimodular cocompact groups cannot fix any point at infinity except in the Euclidean factor; this fact is needed for the study of CAT(0) lattices. Various fixed point results are derived as illustrations. © 2012 Springer-Verlag Berlin Heidelberg.
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Caprace, P.-E., & Monod, N. (2013). Fixed points and amenability in non-positive curvature. Mathematische Annalen, 356(4), 1303-1337. https://doi.org/10.1007/s00208-012-0879-9 (Original work published 2013)