Benjamin Fine - Topics in Infinite Group Theory: Nielsen Methods, Covering Spaces, and Hyperbolic Groups (De Gruyter STEM)
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De Gruyter STEM
ISBN 9783110673340
e-ISBN (PDF) 9783110673371
e-ISBN (EPUB) 9783110673401
Bibliographic information published by the Deutsche Nationalbibliothek
The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de.
2021 Walter de Gruyter GmbH, Berlin/Boston
Mathematics Subject Classification 2020: 08A50, 20EXX, 20FXX, 20H10, 57MXX, 11AXX, 11B37, 11B39, 11D09, 11D25,
Combinatorial group theory is the part of group theory which is presented using generating sets and systems of defining relations. The main principle is the construction of free objects, here called free groups. Their universal property implies that each group is a factor group of a free group.
We start with some notations. Let G be a group. If H is a subgroup of G, then we write H, and if H is a normal subgroup of G, then we write HG . Let XG . We denote the subgroup of G generated by X with X and call X a generating system of X . We have
and we call a formal expression
an X-word, a word in X or simply a word if X is fixed. A word is just a sequence of elements from XX1 where X1={x1xX} . Each word w represents in G an element g:
The neutral element 1 is represented by the trivial or empty word. The inverse word of
is
A word w=x11x22xnn is called (freely) reduced if i+i+10 whenever xi=xi+1 (for i=1,2,,n1 ). We also write w as
Two words v and w are called freely equivalent if one can be transformed into the other by inserting and deleting peaks xx , xX , =1 . This we also denote as vw . A group G is called an -generated group if G can be generated by a set of cardinality . In this sense, G is finitely generated if G can be generated by a finite set. The least number of elements needed to generate G is often called the rank of G and is denoted by rk(G) . If XG , then we write X or XN to denote the normal closure of X in G, that is,
If wG is a word in X then we often write
to indicate that w is a word in the letters x1,x2,,xn of X.
Let X be a set, F a group and i:XF an injective map. The group F, more concrete the pair (F,i) , is called free on X, if it satisfies the following universal property: For each group H and each map f:XH there exists a uniquely determined homomorphism, :FH with f=i , that is, the following diagram (see ) commutes.
Figure 1.1 Commuting diagram.
In addition, we define the trivial group to be free on the empty set . Most commonly we consider X as a subset of F with i as the inclusion. In this case we call X a free generating system or basis of F, we have X= if F={1} .
The infinite cyclic group (Z,+) is free on {1}.
Consider the diagram as in . By definition we have uniquely determined homomorphisms :F1F2 and :F2F1 such that i1=i2 and i2=i1 . Also by definition there is only one homomorphism f:F1F1 with i1=fi1 , hence f==idF1 . Analogously =idF2 .
Figure 1.2 Isomorphism diagram.
We now discuss the existence of a free group on a given set X, that is, we construct a free group on X. Let X be a set. Let X1={x1xX} be another set with XX1= and a bijection xx1 from X to X1 . Let M(X) be the set of all finite sequences (x1,x2,,xn) with xiXX1 , n0 , where we have the empty sequence for n=0 . We define a multiplication on M(X) in succession notation:
This multiplication is associative with neutral element 1, the empty sequence. The map
is injective. We identify (x) with x. With this we may write each element wM(X) uniquely as
where we identify xi+1 with xi . The set M(X) is called the free monoid on XX1 . We call the elements of XX1 letters.
As above, the word w=x11x22xnn is called reduced if xixi+1 or xi=xi+1 but i+i+10 for 1i. The empty word is also reduced. If a word wM(X) is not reduced then we may delete a peak xx , =1 , to get a word wM(X) and we get w back if we insert xx again. This way we get an equivalence relation on M(X) via the definition: If w,vM(X) then wv if w=v in M(X) or w is freely equivalent to v, that is, we get v from w by deleting and inserting peaks. In fact, this is a congruence, that is, if ww then uwvuwv for all u,vM(X) , and if ww , uu then uwuw . Certainly for each w there exists a reduced w with ww .
These facts are straightforward to show. Hence, the multiplication on M(X) implies a multiplication on F(X):=M(X)/ , the set of equivalence classes of M(X) with respect to , via
where [w] denotes the class of w. This multiplication is associative by transmission to the quotient and has a neutral element 1, the class of the empty word. F(X) is a group because if w=x11x22xnn and w=xnnxn1n1x11 then ww1 . The inclusion i:XM(X) induces an inclusion j:XF(X) , and F(X)=j(X) . If we consider, as usual, X as a subset of F(X) , then F(X)=X .
One also may deduce this from the calculation in [].
Let G be a group and f:XG a map. We extend this map via
to a homomorphism f:M(X)G . If xi=xi+1 and i=i+1 , then
Hence, if ww then w and w have the same image in G. Hence, f induces a map :F(X)G via ([w])=f(w) , is a homomorphism with j=f . Because F(X)=j(X) there is only one homomorphism from F(X) to G with fixed values on j(X) .
Let G be a group with G=X . Then G is isomorphic to a factor group of F(X) .
As in the proof of Theorem we may extend the inclusion i:XG to a homomorphism f:F(X)G . Since G=X , f is surjective. Therefore GF(X)/ker(f) by the homomorphism theorem for groups.
Let G be a group and GF(X)/N , F(X) free on X, NF(X) .
An element rN is called a relator (relative to X and G).
A system R
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