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Een n and some X. of e n = e. and g i fixed g e = ~1. be an e l e m e n t g whieh ON GRAPHS and edge if and only Proof. least of e s > ge e LEMMA. vertex g ACTING of union X. of X X. or for e a c h E path" ''' of vertex w of E w,ell,vl,e22,... is a chain is all of of p r o p e r G. 3. Consider any g e o d e s i c in X, v0,e~l ..... e ~ n , v n 29 FIXED POINTS Notice that either G : G v0 then there e~l... hg exist shifts geodesic el, has contradicting Gv0 g ... ~ Gvi here, the subgroup this m e a n s t h a t the directed system Suppose Then G ~ ...

1 the kernel and further, N 38 FUNDAMENTAL II N\F(G,T) = X 1 + ~(X,T') in a n a t u r a l + ~(G,T) Let us n o t e we h a v e normal This a surjection says that a surjection generated 2. maximal of Y will group on a t r e e and the k e r n e l and of Groups groups. then there is the of X. is t h e vertex X of the v e r t i c e s a vertex way precise DEFINITION. will of subgroup X. is of G 0 that we s h a l l consist following a family (tv:~V) e that specialization; theoretic notion but we s h a l l not the c o n n e c t i o n .

1 stabilizer. connected holds (Reidemeister only if it act s f r e e l y G = ~(G\X,T) Further, graph of groups THEOREM is a~ain Proof. (Schreier free. Let G [27]). A group subtree rank H (G:H)( be free on a set set E. G X say. T freely on some at the is free if and In this event, of A subgroup rank G of loops with edge group of G\X. D of this. If m o r e o v e r = = Y. 2 acts freely on ~(Y,T)kF(Y,T) it acts there theorems. [32]). i E, and H of a free ~roup (G:H) rank G then - 1) + 1. and let Then the maximal are finite G Y be the bouquet subtree T of Y is 37 THE TRIVIAL just the u n i q u e v e r t e x of CASE Y, and G = n(Y,T) so acts f r e e l y on the tree X = F(Y,T).

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