J. W. CANNON, W. J. FLOYD, M. A. GRAYSON AND W. P. THURSTON SOLVGROUPS ARE NOT ALMOST CONVEX* ABSTRACT.We show that no cocompact discrete group based on solvgeometry, Sol, is almost convex. This reflects the geometry of Sol, and implies that the Cayley graph of a cocompact discrete group based on Sol cannot be efficiently constructed by finitely many local replacement rules. 1. INTRODUCTION We show that no cocompact discrete group based on solvgeometry, Sol, is almost convex. Almost-convexity is a metric property satisfied by all cocompact hyperbolic groups, all Euclidean groups, all free products with amalgamation of finite groups, all HNN extensions of finite groups, and all small cancellation groups [1]. Intuition suggests that it should be satisfied by those cocompact groups based on geometries having unique shortest geodesics and convex metric balls. Such is certainly true in geometries having sectional curvatures ~< 0. Therefore the property is likely to apply to braid groups, mapping class groups, complex hyperbolic groups, groups of higher rank symmetric spaces whose factors have convex metric balls, etc. It is likely to apply to nilgroups as well, whose metric balls, though not convex, are almost convex. The property of almost-convexity is necessary and sufficient in order that the Cayley graph of a group be efficiently constructible by means of finitely many local replacements rules [2]. There is some hope that many groups which are not almost convex can nevertheless be studied by means of such local replacement rules if one is willing to embed them in larger groups or graphs which are almost convex. In particular, solvgeometry So! embeds in complex hyperbolic space. Local replacement rules seem closely related to rational growth functions in groups. Nevertheless, a number of cocompact groups based on solvgeometry have rational growth functions. This was first claimed in Chapter 4 of M. A. Grayson's thesis [3], but there was an error in his proof. The error was discovered by W. Parry and was corrected in [4]. Our result shows how clearly the combinatorial structure of a geometric group mirrors the properties of the geometry on which it is based: shortest geodesics in Sol are highly nonunique. Our result has significance in the study of 3-manifolds and their groups. W. P. Thurston has conjectured that each *This research was supported in part by NSF Research Grants. Geometriae Dedicata 31: 291-300, 1989. © 1989 Kluwer Academic Publishers. Printed in the Netherlands.