On the classification of CAT(0) structures for the 4-string braid group

Abstract

This paper is concerned with the class of so-called CAT(0) groups, namely, those groups that admit a geometric (i.e., properly discontinuous, co-compact, and isometric) action on some CAT(0) space. More precisely, we are interested in knowing to what extent it is feasible to classify the geometric CAT(0) actions of a given group (up to, say, equivariant homothety of the space). A notable example of such a classification is the flat torus theorem, which implies that the minimal geometric CAT(0) actions of the free abelian group Z (n ≥ 1) are precisely the free actions by translations of Euclidean space E. Typically, however, a given group will have uncountably many nonequivalent actions, making any chance of a complete classification rather slim. It is therefore reasonable to consider, for example, only those actions of a group on a space of minimal possible dimension—namely, the geometric dimension of the group. Thus, the geometric actions of the free group Fn, n ≥ 2, on 1-dimensional CAT(0) spaces (R-trees) are classified by the compact metric graphs of Euler characteristic 1− n. Even so, the variety of, say, the 2-dimensional CAT(0) structures for closed surface groups would seem to be vast, with many of these structures being nonplanar (see Section 5.1). By contrast, we are able to cite examples of CAT(0) groups of geometric dimension 2 that have no 2-dimensional CAT(0) structure [7; 3]. Other results in a similar vein are to be found in [2], [12], and [11]. Consider the n-string braid group Bn defined by the following presentation: Bn = 〈a1, a2 , . . . , an−1 | aiai+1ai = ai+1aiai+1 for i = 1, . . . , n− 2; aiaj = aj ai if |i − j | ≥ 2〉. The braid group Bn has geometric dimension n−1, and it is known to be CAT(0) (with a structure in dimension n−1) if n ≤ 5 [5; 4; 6]. The question of existence of a CAT(0) structure is open for n ≥ 6. We note that each braid group has infinite cyclic center (see [9]). The group B3 acts geometrically on the product T × R, where T is a trivalent tree, and up to an independant homothety of each factor of the product this is the unique minimal CAT(0) structure. This fact was used by Hanham in his thesis [12]. We give a proof here (see Theorem 4) that simply reduces the question to