# Rational subgroups of biautomatic groups

@article{Gersten1991RationalSO, title={Rational subgroups of biautomatic groups}, author={S. M. Gersten and H. B. Short}, journal={Annals of Mathematics}, year={1991}, volume={134}, pages={125-158} }

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

## 182 Citations

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## References

SHOWING 1-10 OF 29 REFERENCES

Reducible Diagrams and Equations Over Groups

- Mathematics
- 1987

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,…

Combings of Groups

- Mathematics
- 1992

We use combings to prove a theorem a la Rips. Applying it to hyperbolic groups one recovers Rips’ Theorem. When applied to automatic groups the theorem yields, via a theorem of Ken Brown, a proof…

Discrete Groups of Motions

- MathematicsCanadian Journal of Mathematics
- 1960

This paper deals with the discrete groups of rigid motions of the hyperbolic plane. It is known (12) that the finitely generated, orientation-preserving groups have the following presentations:…

Lectures on three-manifold topology

- Mathematics
- 1980

Loop theorem-sphere theorem: The Tower Construction Connected sums 2-manifolds embedded in 3-manifolds Hierarchies Three-manifold groups Seifert fibered manifolds Peripheral structure Essential…

Small cancellation theory and automatic groups: Part II

- Mathematics
- 1991

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)…

Real length functions in groups

- Mathematics
- 1972

This paper is a study of the structure of a group G equipped with a 'length' function from G to the nonnegative real numbers. The properties that we require this function to satisfy are derived from…

Finite Automata and Their Decision Problems

- Computer ScienceIBM J. Res. Dev.
- 1959

Finite automata are considered as instruments for classifying finite tapes as well as generalizations of the notion of an automaton are introduced and their relation to the classical automata is determined.

Manifolds of Nonpositive Curvature

- Mathematics
- 1985

Preface.-Introduction.-Lectures on Manifolds of Nonpositive Curvature.-Simply Connected Manifolds of Nonpositive Curvature.-Groups of Isometries.-Finiteness theorems.-Strong Rigidity of Locally…

Dendrology of Groups: An Introduction

- Biology
- 1987

The study of group actions on “generalized trees” or “ℝ-trees” has recently been attracting the attention of mathematicians in several different fields and has been developed further by the above-mentioned people and also by H. Culler, H. Gillet, M. Rimlinger and J. Stallings.