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X E X is a h o m o m o r p h i s m smallest to g e n e r a t e G. 3. 8, free sending (Gi)e kx, m indexed be the Then there kx, m (Gi)~ required F ÷ G freely K i. kx,nx is a m o n o m o r p h i s m be the sending x8 1 ~ m s nx, Y. e:G + K generating an e x p r e s s i o n on follows fact because i • I. Proof. 4 THEOREM rank(A HB) (Grushko = rank A Proof. Neumann + rank B) ~ same [43]). For any s r o u p s A,B, B. rank A r a n k as 8:F + A H B . = B. 3, + rank rank Let so t h e r e Hence : B. FB B.

So we m a y For f > g*. g*. f on lies g is c o m p a r a b l e which ~ e ~ g e > f*, comparable set e is c o m p a r a b l e construct and f Now ordered e E, relation element equal way. partially if e i t h e r suppose GRAPHS E. notation, transitive, e,f,g e ~ f ON in t h i s any an e q u i v a l e n c e is, of that of (double) arise be a n o n e m p t y such one Define e,f can ACTING (f*,e] u [g*,f) 33 TREES AND PARTIAL So e covers g*. §9 ORDERS It is n o w clear that ~ is an e q u i v a l e n c e relation.

Then the p a r t i a l for any in E, f-i < e I if and only if f* < e in E. (iii) we now h a v e (Dunwoody of an d the E, a n d the to an a c t i o n e G (E, ~, *) ordered) G on {g e G I ge < e X or acts then in E, under tree on E ge* < e}. a way D ~ s u c h that to e x a c t l y [e,f] one is finite. X. r e s p e c t i n $ the p a r t i a l the a c t i o n in such be a n o n e m p t y involution interval set of a c e r t a i n involution, of Let is c o m p a r a b l e a group e,f order following.