M.Sc Thesis

M.Sc StudentWeiss Uri
SubjectSmall Concellation Techniques in Automatic Groups
DepartmentDepartment 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