A characterisation of virtually free groups by Gilman R.H., Hermiller S., Holt D.F.

By Gilman R.H., Hermiller S., Holt D.F.

We turn out finitely generated crew G is nearly unfastened if and provided that there exists a producing set for G and к > zero such that each one k-locally geodesic phrases with admire to that producing set are geodesic.

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F o r t h e group S 2 n the and 0- as w e l l as t h e i r c o m m u t a t o r s u b g r o u p s a r e 2n 2n t d e f i n e d t o be of d e p t h one i n S . The i n t e r s e c t i o n o f O Z n a n d 02n i s a d i r e c t p r o d u c t o f Z w i t h S 2 ( n - l ) , where Z 2 d e n o t e s a in S 2n 2 c y c l i c g r o u p o f o r d e r t w o . T h i s i n t e r s e c t i o n , as w e l l as i t s s u b g r o u p s 0' S2(n-l) s u b g r o u p , a r e d e f i n e d t o b e of d e p t h two i n S .

On t h e o t h e r h a n d , t h e r e a r e some c a s e s i n w h i c h two d i s t i n c t s y m b o l s d e n o t e t h e same c h a r a c t e r f o r o n e p a r t i c u l a r v a l u e o f n b u t n o t f o r any l a r g e r v a l u e s o f n . Second, not a l l p o s s i b l e a r r a y s occur; f o r i n s t a n c e t h e r e i s no c h a r a c t e r ( o f l e v e l t w o ) whose a r r a y h a s 0 1 1 i n t h e t o p row a n d 1 0 1 i n t h e b o t t o m r o w . A s a m a t t e r of f a c t , 44 FRAME AND KIJDVALIS we d o n o t a s y e t know ( e x c e p t i n a p o s t hoc way) which a r r a y s a c t u a l l y d o o c c u r o r how t o a s s i g n t h e c o n s t a n t f r a c t i o n t o t h e o n e s t h a t d o o c c u r t o c o m p l e t e t h e symbol.

T h i s i n t e r s e c t i o n , as w e l l as i t s s u b g r o u p s 0' S2(n-l) s u b g r o u p , a r e d e f i n e d t o b e of d e p t h two i n S . Groups o f g r e a t e r d e p t h i n S are t h e n d e f i n e d i n d u c t i v e l y , s o t h a t a subgroup o f d e p t h k t 1 I n S 2 n i s , e x c e p t p o s s i b l y for a f a c t o r Z2, a s u b g r o u p of d e p t h k-1 i n S 2 ( n - l ) . A l t h o u g h w e d o n o t as y e t h a v e a p r o o f o f t h i s , e x a m p l e s seem t o b e a r o u t t h e f o l l o w i n g c o n j e c t u r e : CONJECTURE 8 .

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