One relator quotients of surface groups

A surface group is the fundamental group of an orientable or a non-orientable surface. It is well known that if S is an orientable surface of genus k ~ 2, then the corresponding surface group also has genus k ~ 2 and it has a one-relator presentation of the form 1f1(Sk) = (all bll ···, ak, bk : [aI, bl ]··· [ak' bk] = 1). The thesis consists of research work on one-relator surface groups. One-relator surface groups are natural generalizations of one-relator groups. These groups are obtained as quotients of the fundamental group of an orientable surface by the normal closure of a single element. We are interested in the quotients of the surface group 1f1(Sk). Let R E 1f1(Sk) be a single element, then the quotient 1f1(Sk)/((R)) is the group G = (aI, bl , ... , ak, bk : [aI, bl ]··· [ak' bkl = R = 1), where the relator R is cyclically reduced word on the generators of G. We define Magnus subgroup for a one-relator surface group and prove that they are cyclonormal. This is a natural analogue of a Theorem of Bagherzadeh about one-relator groups. If we have two Magnus subgroups MA and MB of G, then we shall describe the structure of their intersection MA nc MB • We will discuss the intersection of conjugates of Magnus subgroups of G. These results are analogues of two Theorems of Collins about one-relator groups. We will generalize Weinbaum's result about one-relator groups to one-relator surface groups and give necessary and sufficient conditions that if R = UV, then U = 1 in G. An algebraic proof of the Freiheitssatz is given for one-relator surface groups. This result is a natural generalization of Magnus's Freiheitssatz for one-relator groups.EThOS - Electronic Theses Online ServiceGBUnited Kingdo