Central Quotients of Biautomatic Groups

The quotient of a biautomatic group by a subgroup of the center is shown to be biautomatic. The main tool used is the Neumann-Shapiro triangulation of S, associated to a biautomatic structure on Zn. As an application, direct factors of biautomatic groups are shown to be biautomatic. Biautomatic groups form a wide class of finitely presented groups with interesting geometric and computational properties. These groups include all word hyperbolic groups, all fundamental groups of finite volume Euclidean and hyperbolic orbifolds, and all braid groups [E. . . ]. A biautomatic group satisfies a quadratic isoperimetric inequality, has a word problem solvable in quadratic time, and has a solvable conjugacy problem. The class of biautomatic groups has several interesting closure properties. For instance, the centralizer of a finite subset of a biautomatic group is biautomatic, as is the center of the whole group [GS]. Also, biautomatic groups are closed under direct products [E. . . ]. The theory of biautomatic groups is briefly reviewed below. We present a technique for putting biautomatic structures on central quotients of biautomatic groups: Theorem A. Let G be a biautomatic group, and let C be a subgroup of ZG, the center. Then G/C is biautomatic. This result has several applications. Our first application answers a question posed by Gersten and Short [GS, cf. proposition 4.7]: Theorem B. Direct factors of biautomatic groups are biautomatic. Proof. Suppose G × H is biautomatic. The centralizer of H is G × ZH, and this is a biautomatic group by [GS, corollary 4.4]. Then ZH is a subgroup of the center of G×ZH, so by theorem A, G× ZH/ZH = G is biautomatic. ⋄ Several recent discoveries have pointed to the useful concept of poison subgroups. For instance, the group Z is poison to word hyperbolic groups: if a group contains a Z subgroup it cannot be word hyperbolic. A wider class of subgroups poison to word hyperbolicity are those which have an infinite index central Z subgroup [CDP, corollaire 7.2]. Our next theorem says that for biautomatic groups, this class of poison subgroups completely collapses to Z: The author was partially supported by NSF grant # DMS-9204331 This preprint is available from the Magnus archive on the American Mathematical Society’s e-MATH host computer. Send the message “HELP” to [email protected]