We prove that the cohomological dimension of the Torelli group for a closed, connected, orientable surface of genus g ≥ 2 g \geq 2 is equal to 3 g − 5 3g-5 . This answers a question of Mess, who proved the lower bound and settled the case of g = 2 g=2 . We also find the cohomological dimension of the Johnson kernel (the subgroup of the Torelli group generated by Dehn twists about separating curves) to be 2 g − 3 2g-3 . For g ≥ 2 g \geq 2 , we prove that the top dimensional homology of the Torelli group is infinitely generated. Finally, we give a new proof of the theorem of Mess that gives a precise description of the Torelli group in genus 2. The main tool is a new contractible complex, called the “complex of minimizing cycles”, on which the Torelli group acts.

Abstract. Let Tn be the kernel of the natural map Out(Fn) → GLn(Z). We use combinatorial Morse theory to prove that Tn has an Eilenberg-MacLane space which is (2n − 4)-dimensional and that H 2n−4 (Tn, Z) is not finitely generated (n ≥ 3). In particular, this recovers the result of Krstić-McCool that T 3 is not finitely presented. We also give a new proof of the fact, due to Magnus, that Tn is finitely generated.

Let G(O S ) be an S-arithmetic subgroup of a connected, absolutely almost simple linear algebraic group G over a global function field K. We show that the sum of local ranks of G determines the homological finiteness properties of G(O S ) provided the K-rank of G is 1. This shows that the general upper bound for the finiteness length of G(O S ) established in an earlier paper is sharp in this case.The geometric analysis underlying our result determines the conectivity properties of horospheres in thick Euclidean buildings.

Abstract. We prove the result from the title using the geometry of Euclidean buildings. Mathematics Subject Classification (2000). 20F65.Keywords. Euclidean buildings, finiteness properties, geometric group theory. IntroductionLittle is known about the finiteness properties of SL n .ZOEt/ for arbitrary n.In 1959 Nagao proved that if k is a field then SL 2 .kOEt/ is a free product with amalgamation [Na]. It follows from his description that SL 2 .ZOEt/ and its abelianization are not finitely generated.In 1977 Suslin proved that when n 3; SL n .ZOEt/ is finitely generated by elementary matrices [Su]. It follows that H 1 .SL n .ZOEt/; Z/ is trivial when n 3.More recent, Krstić and McCool proved in [Kr-Mc] that SL 3 .ZOEt/ is not finitely presented.In this paper we provide a generalization of the results of Nagao and Krstić-McCool mentioned above for the groups SL n .ZOEt/.Recall that a group is of type FP m if there exists a projective resolution of Z as the trivial Z modulewhere each P i is a finitely generated, projective Z module. In particular, Theorem 1 implies that there is no K.SL n .ZOEt/; 1/ with finite .n 1/-skeleton, where K.G; 1/ is the Eilenberg-Mac Lane space for G. . Here we will examine the action of SL n .ZOEt/ on the locally infinite Euclidean building for SL n .Q..t 1 ///. In Section 2 we will show that the infinite groups that arise as cell stabilizers for this action are of type FP m for all m, which is a technical condition that is needed for our application of Brown's criterion.In Section 3 we will demonstrate the existence of a family of diagonal matrices that will imply the existence of a "nice" isometrically embedded codimension 1 Euclidean space in the building for SL n .Q..t 1 ///. In [Bu-Wo1] analogous families of diagonal matrices were constructed using some standard results from the theory of algebraic groups over locally compact fields. Because Q..t 1 // is not locally compact, our treatment in Section 3 is quite a bit more hands on.Section 4 contains the main body of our proof. We use translates of portions of the codimension 1 Euclidean subspace found in Section 3 to construct spheres in the Euclidean building for SL n .Q..t 1 /// (also of codimension 1). These spheres will lie "near" an orbit of SL n .ZOEt/, but will be nonzero in the homology of cells "not as near" the same SL n .ZOEt/ orbit. Theorem 1 will then follow from Brown's criterion. Background material.Our proof relies heavily on the geometry of the Euclidean and spherical buildings for SL n .Q..t 1 ///. A good source of information for the former topic is Chapter 6 of [Br2]. For the latter, we recommend Chapter 5 of [Ti]. StabilizersLemma 2. If X is the Euclidean building for SL n .Q..t 1 ///, then the SL n .ZOEt/ stabilizers of cells in X are FP m for all m.Proof. Let x 0 2 X be the vertex stabilized by SL n .QOEOEt