Files in this item
Files  Description  Format 

application/pdf 8203393.pdf (2MB)  (no description provided) 
Description
Title:  Solutions of Quadratic Equations in Small Cancellation Groups 
Author(s):  Anderson, Claude Wilson, III 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  The main results are classifications of the solutions of quadratic equations in small cancellation quotients of free groups and free products. Paul E. Schupp has obtained similar results for a restricted class of quadratic equations (those without constants). Some of the methods used here were previously employed by Leo P. Comerford and Charles C. Edmunds in their study of algorithmic problems on the solutions of quadratic equations. The principal tools are the construction of cancellation diagrams on compact surfaces with boundary and an analysis of the coinitial graphs of quadratic words. Let F = and H = be free groups. The generators of F and their inverses are called variables, while the generators of H and their inverses are constants. A word W of F*H is quadratic if each variable which occurs in W occurs exactly twice, with exponent +1 or 1 in each occurrence. The words (alpha)(,1)h(,1)(alpha)(,2)(alpha)(,1)('1)(alpha)(,2)('1), (alpha)(,1)('2)(alpha)(,2)('2)(alpha)(,3)('2), and (alpha)(,1)h(,1)(alpha)(,2)h(,2)(alpha)(,1)h(,1)(alpha)(,2) are quadratic. If (alpha)(,1),...,(alpha)(,n) and h(,1),...,h(,k) are the variables and constants, respectively, occurring in W, we write W((alpha)(,1),...,(alpha)(,n); h(,1),...,h(,k)). Let R be a symmetrized subset of H, and let N be the normal closure of R in H. Let G = H/N and let (phi):H (>) G be the natural map, so that words of H define elements of G. We look at solutions of W = 1 in G. The tuple (a(,1),...,a(,n)) of words of H is a solution of the equation W((alpha)(,1),...,(alpha)(,n); h(,1),...,h(,k)) in G if the word W(a(,1),...,a(,n); h(,1),...,h(,k)) of H defines the identity element of G. Much is known about the nature of solutions of quadratic equations in free groups. A free solution of W = 1 in G is induced by a solution of the same equation in H. The solution (a(,1),...,a(,n)) is free if there are words (z(,1),...,z(,n)) of H such that (phi)(z(,i)) = (phi)(a(,i)) for all i and W(z(,1),...,z(,n); h(,1),...,h(,k)) = 1 in H. If we can show that all solutions of W = 1 in a given group G are free, then we can use the classification of solutions in the free group to classify the solutions in G. If R satisfies sufficiently strong small cancellation conditions, then all solutions of W = 1 in G are free. We first show that a nonfree solution induces a reduced Rdiagram on a compact surface with boundary. Then we use a combinatorial argument to show that the small cancellation conditions preclude the existence of such a diagram. For this argument we need a lower bound on the Euler characteristics of the surfaces involved; this is where we employ the coinitial graph. We also obtain similar results when H is a free product. The constants in the equations are arbitrary nontrivial elements of free factors. 
Issue Date:  1981 
Type:  Text 
Language:  English 
Description:  82 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1981. 
URI:  http://hdl.handle.net/2142/68184 
Other Identifier(s):  (UMI)AAI8203393 
Date Available in IDEALS:  20141214 
Date Deposited:  1981 
This item appears in the following Collection(s)

Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois