S. M. Gersten

Rational subgroups of biautomatic groups

Centralizers of finite subsets in biautomatic groups are them- selves biautomatic. Every polycyclic subgroup of a biautomatic group is abelian by finite.

Automatic groups and amalgams

Abstract The objectives of this paper are twofold. The first is to provide a self-contained introduction to the theory of automatic and asynchronously automatic groups, which were invented a few years ago by J.W. Cannon, D.B.A. Epstein, D.F. Holt, M.S. Paterson and W.P. Thurston. The second objective is to prove a number of new results about the construction of new automatic and asynchronously automatic groups from old ones by means of amalgamated products.

Reducible Diagrams and Equations Over Groups

Diagrammatic reducibility is related to the solution of equations over groups. Sufficient conditions for the reducibility of all spherical diagrams are given, unifying and generalizing work of Adian, Remmers, Lyndon, and Sieradski. Hyperbolic 2-complexes are defined and the word problem is solved for their fundamental groups.

A presentation for the special automorphism group of a free group

A presentation is given for SAn, the group of automorphisms of determinant 1 of a free group Fn of rank n. The canonical isomorphisms H2(An,Z)∋H2(SAn,Z)∋K2(Z) are established for n ≥ 5, where An is the full group of automorphisms of Fn.

Quadratic Divergence of Geodesics in CAT (0) Spaces.

A finite CAT(0) 2-complexX is produced whose universal cover possesses two geodesic rays which diverge quadratically and such that no pair of rays diverges faster than quadratically. This example shows that an aphorism in Riemannian geometry, that predicts that in nonpositive curvature nonasymptotic geodesic rays either diverge exponentially or diverge linearly, does not hold in the setting of CAT(0) complexes. The fundamental group ofX is that of a compact Riemannian manifold with totally geodesic boundary and nonpositive sectional curvature.

Small cancellation theory and automatic groups: Part II

The main results obtained in Part I can be rephrased in the language of root systems and Euclidean planar tessellations as saying that finite A1 x A1 and A 2 piecewise Euclidean (abbreviated PE) complexes of non-positive curvature have automatic fundamental groups. (A 2-dimensional complex is a A 1 • A1, respectively A2, PE complex if every 2-cell can be identified with a unit square, resp., equilateral triangle in the Euclidean plane, in such a way that the induced metrics agree on intersections. Such a complex has non-positive curvature if the link of each vertex contains no circuits without backtracking of length less than 2 n). Here we prove using similar techniques that finite 2-complexes modelled on the other two root systems, B 2 and G2, corresponding to tessellations by 45 ~ right angle triangles and by 30 ~ 60 ~ right angle triangles, also have automatic fundamental groups (Theorems 2.3 and 3.2). To be precise,

Cohomological lower bounds for isoperimetric functions on groups

Abstract If the finitely presented group G splits over the finitely presented sub-group C, then classes are constructed in H 2 (∞) (G) which reflect the splitting and which serve as lower bounds for isoperimetric functions for G. It is proved that H 2 (∞) (G)=0 for all word hyperbolic groups G. A converse is obtained for the combination theorem for hyperbolic groups of Bestvina–Feighn. The Mayer–Vietoris exact sequence for l ∞ -cohomology associated to a splitting of a group is established. Metabolic groups are introduced as finitely presented groups G such that H 2 (∞) (G, A)=0 for all normed abelian coefficient groups A and such groups G are shown to be characterized by possessing “thin” combings.

Fixed points of automorphisms of free groups

Abstract The set of fixed points of an automorphism of a finitely generated free group is a finitely generated group, settling a conjecture of G. P. Scotts.

BOUNDED COCYCLES AND COMBINGS OF GROUPS

We adopt the notion of combability of groups defined in [12]. An example is given of a (bi-)combable group which is not residually finite. Two of the eight 3-dimensional geometries, and ℍ2×ℝ, are quasiisometric. Seifert fibred manifolds over hyperbolic orbifolds have bicombable fundamental groups. Every combable group satisfies an exponential isoperimetric inequality.

Divergence in 3-manifold groups

The divergence of the fundamental group of compact irreducible 3-manifolds satisfying Thurstons geometrization conjecture is calculated. For every closed Haken 3-manifold group, the divergence is either linear, quadratic or exponential, where quadratic divergence occurs precisely for graph manifolds and exponential divergence occurs when a geometric piece has hyperbolic geometry. An example is given of a closed 3-manifoldN with a Riemannian metric of nonpositive curvature such that the divergence is quadratic and such that there are two geodesic rays in the universal cover∼N whose divergence is precisely quadratic, settling in the negative a question of Gromovs.