By Benjamin Fine, Gerhard Rosenberger

A survey of one-relator items of cyclics or teams with a unmarried defining relation, extending the algebraic research of Fuchsian teams to the extra normal context of one-relator items and comparable workforce theoretical issues. It presents a self-contained account of yes ordinary generalizations of discrete teams.

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**Example text**

M}, {h~(h’), h~-¢(h~)} < {g~(a’), g~-~(a’)} fails to hold. ,g,~} which is shorter. If G is countable then every finite system can be carried by a Nielsen transformation into a minimal system. In general, as already mentioned, for a given finite system, a suitable order can always be chosen such that this finite system can be carried by a Nielsen transformation into a minimal system. The Nielsen reduction method in G now refers to Nielsen transformations from given systems to shorter systems and the resulting investigation of minimal systems.

What ties these all together is the realtionship between the group structure and the geometric structure of its Cayley graph. Geometric group theory will appear occasionally so in this section and the next two sections we survey the main definitions and ideas, looking first at hyperbolic groups. MaxDehn in his pioneering work on combinatorial group theory [De 1] introduced the following three fundamental group decision problems. (1) Word Problem: Suppose G is a group given by a finite presentation.

Rewrite rules need not shorten the length of a word. However we do get the following interesting tie with both free groups and hyperbohc groups. 2. Let G be a ~nitely generated group with a finite, length reducing, complete rewriting system. Then this system gives a Dehn algorithm and hence G must be hyperbolic. ~r~her G is virtually free, that is has a free subgroupof finite index. Another remarkable result along these lines was obtained by Muller and Schupp [Mu-Sch]. Let N(F) be the set of words W E F with #(W) -- 1.