We show that the finiteness length of an S-arithmetic subgroup Γ in a noncommutative isotropic absolutely almost simple group G over a global function field is one less than the sum of the local ranks of G taken over the places in S. This determines the finiteness properties for arithmetic subgroups in isotropic reductive groups, confirming the conjectured finiteness properties for this class of groups.Our main tool is Behr-Harder reduction theory which we recast in terms of the metric structure of euclidean buildings. for Mathematics (Bonn) for their hospitality within the trimester programs "Rigidity" and "Algebra and number theory". We also gratefully acknowledge the hospitality of the Mathematisches Forschungsinstitut Oberwolfach during several workshops and their RiP program. The DFG provided appreciated financial and structural support via SFB 701 at Bielefeld University and via GR 2077/5 and GR 2077/7 at Darmstadt.We are also indebted to Bernd Schulz for explaining his thesis. The participants of the Bielefeld seminar on reduction theory in summer 2010 (in particular Herbert Abels, Werner Hoffmann, Gregory Margulis, and Andrei Rapinchuk) provided valuable expertise. We thank

Abstract. Based on the second author's thesis [33], in this article we provide a uniform treatment of abstract involutions of algebraic groups and of Kac-Moody groups using twin buildings, RGD systems, and twisted involutions of Coxeter groups. Notably we simultaneously generalize the double coset decompositions established in [32] and [49] for algebraic groups and in [39] for certain Kac-Moody groups, we analyze the filtration studied in [24] in the context of arbitrary involutions, and we answer a structural question on the combinatorics of involutions of twin buildings raised in [7].

Abstract. Let G be a split real Kac-Moody group of arbitrary type and let K be its maximal compact subgroup, i.e. the subgroup of elements fixed by a Cartan-Chevalley involution of G. We construct non-trivial spin covers of K, thus confirming a conjecture by Damour and Hillmann. For irreducible simply-laced diagrams and for all spherical diagrams these spin covers are two-fold central extensions of K. For more complicated irreducible diagrams these spin covers are central extensions by a finite 2-group of possibly larger cardinality. Our construction is amalgam-theoretic and makes use of the generalised spin representations of maximal compact subalgebras of split real Kac-Moody algebras studied by Hainke, Levy and the third author. Our spin covers contain what we call spin-extended Weyl groups which admit a presentation by generators and relations obtained from the one for extended Weyl groups by relaxing the condition on the generators so that only their eighth powers are required to be trivial.

A lemma of Tits establishes a connection between the simple connectivity of an incidence geometry and the universal completion of an amalgam induced by a sufficiently transitive group of automorphisms of that geometry. In the present paper, we generalize this lemma to intransitive geometries, thus opening the door for numerous applications. We treat ourselves some amalgams related to intransitive actions of finite orthogonal groups, as a first class of examples. 2000 Mathematics Subject Classification 20E06, 05E20, 05E25, 51A05.