M.Sc Thesis | |

M.Sc Student | Weiss Uri |
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Subject | Small Concellation Techniques in Automatic Groups |

Department | Department of Mathematics |

Supervisor | ASSOCIATE PROFESSOR Arie Juhasz |

Automatic and Biautomatic groups are introduced by Epstein et al. It includes many groups, among them are the finite groups, commutative groups, free-groups, hyperbolic groups, small cancellation groups, braid groups, Coxeter groups, Euclidean Groups and more. Automatic groups have many important properties such as solvable word problem and quadratic isoperimetric inequality. Classical small cancellation groups (namely of types C(3)&T(6), C(4)&T(4) and C(6)) where shown to be biautomatic by S. Gersten and H. Short. We show in this work that biautomaticity of small cancellation groups is retained under non-homogenous small cancellation conditions.

We define a new class of non-homogenous small
cancellation diagrams and name it V(6) diagrams. The main result we give deals
with the corresponding V(6) groups, namely groups with finite presentation for
which the van-Kampen diagrams are V(6) diagrams. We show that V(6) groups are
bi-automatic under the assumption that every piece is of length one. The V(6)
diagrams generalize previously studied homogenous small cancellation diagrams
of types C(4)&T(4) and C(6).

The main part of the
work is devoted to analysis of V(6) diagrams with certain boundary conditions.
We also show that given a finite presentation, biautomaticity is implied by
exhibiting special order on words over the generators. These two key steps
together with a specific word order for V(6) groups with pieces of length one
imply the main result. The main tools used are structure theorem of W(6)
diagrams (which includes the V(6) diagrams) due to A. Juhász and
falsification by k-fellow-traveler techniques due to D. Peifer