No abstract

Abstract. The concept of a gated subset in a metric space is studied, and it is shown that properties of disjoint pairs of gated subsets can be used to investigate projections in Tits buildings.

Abstract. All indecomposable unimodular hermitian lattices in dimensions 14 and 15 over the ring of integers in Q( √ −3) are determined. Precisely one lattice in dimension 14 and two lattices in dimension 15 have minimal norm 3.In 1978 W. Feit [10] classified the unimodular hermitian lattices of dimensions up to 12 over the ring Z[ω] of Eisenstein integers, where ω is a primitive third root of unity. These lattices all have roots, that is, vectors of norm 2. In dimension 13, for the first time a unimodular lattice without roots appears [1,3]. In [2] the unimodular lattices in dimension 13 are completely classified. The root-free lattice turns out to be unique. It has minimal norm 3, and its automorphism group is isomorphic to the group Z 6 × PSp 6 (3) of order 2 10 .3 10 .5.7.13. The remaining lattices all have roots; the rank of the root system is 12 in all cases.In this paper, we classify the unimodular lattices in dimensions 14 and 15. There are exactly 58, respectively 259 classes of indecomposable lattices in these dimensions. Below, we list their root systems and the orders of their automorphism groups. Gram matrices for all lattices are available electronically via www.mathematik.uni-dortmund.de/~scharlauThere is only one root-free unimodular lattice of rank 14, and there are two root-free unimodular lattices of rank 15.The lattices without roots have minimal norm 3; they are extremal as introduced for unimodular Eisenstein lattices in [8], Chapter 10.7. They give rise to 3-modular extremal Z-lattices in twice the dimension, as defined by Quebbemann in [17]. See [8,19,20] for more information on extremal and modular lattices and their relation to modular forms. In this context, the lattices classified in this paper can be considered as complex structures on (extremal) 3-modular lattices. The question for existence, uniqueness, and possibly a full classification of extremal modular lattices has been an ongoing challenge, both computationally and theoretically, after the appearance of the influential paper [17].Let V be a vector space over Q( √ −3) with a positive definite hermitian product (, ). A lattice L in V is a finitely generated Z[ω]-module contained in V such that L contains a basis of V and (x, y) ∈ Z[ω] for all x, y ∈ L. More precisely, one

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Following earlier work of Tits [8], this paper deals with the structure of buildings which are not necessarily thick; that is, possessing panels (faces of codimension 1) which are contained in two chambers, only. To every building A, there is canonically associated a thick building z~ whose Weyl group W(/~) can be considered as a reflection subgroup of the Weyl group W(A) of A. One can reconstruct A from A together with the embedding W(z])% W(A). Conversely, if/~ is any thick building and W any reflection group containing W(A) as a reflection subgroup, there exists a weak building A with Weyl group W and associated thick building z~.