Howson [41praved that the intersection of twa finitely gener- ated subgroups H and ... bound for the number of branch points in the core of V(H) x fl(1
REVISTA MATEMÁTICA
de la
Universidad Complutense de Madrid Volumen 9, número 1: 1996
Qn the Intersection of Finitely Generated Subgroups of Free Groups W.S. JASSIM To A.H.M. Hoare
ABSTRACT. Howson [41praved that the intersection of twa finitely generated subgroups H and 1< of ranks m and it respectively is finitely generated. HepravedthattherankNofffflKisatmast2litit—m—it+l. H. Neumann [8,9] gaye a better bound of 2mn 2it — 2m + 3. Burns [2]further impraved the general upper haund to N < 2mn — 3m 2n + 4 (for m =it). —
—
Imrich [6] gaye shorter proof of Neumann>s result and also Nickolas [10] gaye simple praof for Burn’s result. Servatius [12] gaye graphical praof far Burn’s result. Burns [1] showed that the stranger baund N < mit
— it — m + 2
holds
if 11 or 1< is of finite index in E.
In this paper hold.
it is
shown tbat stronger bound N < mit
—
it
—
m+2
always
In section 1 we gaye basic concepts about free groups, graphs and cayley graphs. 1991 Mathematics Subject Classification: 2aEal Servicio publicaciones Univ. Complutense. Madrid, 1996. 1 would like to thank the referee br lis helpful suggestions.
68
W.S.
.iassim
lix section 2 we showed the main theorem 2.7: “lf 11 and 1< are finitely generated subgroups of a free group E on generators a, b of ranks m, it respectively and if ah vertices un V(H) and r(K) are of degree 2 and 3 only, then the rank N of 11 fl 1< satisfies N < mit — it — m + 2”. In order to prove the main Theorem 2.7, we foflowed and improved the techniques which were used hy Nickolas [10] especially the concept of compatibelty of paths and branch points. By this irnprovement, we could have an upper bound on the number of the compatible branch points in the core of F*(H) >< 1”(le) which is the product of r*(11) and r(le). Therefore we started to know the least nurnber of typing of compatible hranch points of degree 3 only un P(H) as shown un Lemma 2.3. In Lemma 2.4 we showed that if IY(H) has only two types of compatible branch points X1 and 3
b
/
b
A
(4. a>
Q, • 4>
e (5,9)
U
(7,5>
82
W.S.
Jassim
Example 2:
17(H):
17(K)
SS
A
SS
2
A
SS
SS
5,
Core of 17(H) x 17(K):
a
.5 A
(5,3>
e
t2, 4)
(2,0)
u,
b (5;
a
• ¡5>
On tbe Intersection of Finitely Generated...
Example 3: 17(H):
17(K):
Core of 17(H) ~ 17(K):
83
84
W.S.
.Iassim
References [1] Burns, R.G., A note oit free group 14—17.
.
Anier. MatIt. Soc. 23 (1969),
[2] Burns, R.G., Qn tire intersection of finitely gene nated subgroups of free groups. MatIt. Z. 119 (1971), 121-130. [3] Gersten, S.M., Intersection of finitely generated subgnoups of free groups avid resolutions of grapirs. Intrent. MatIt. 71(1983), 567-591. [4] Howson, A.G., Qn tire intersection offinitely gerterated free groups. J. Lon. MatIt. Soc. 29 (1954), 428-434. [5] Irnrich, W., Stibgrotip tireorem on groups. Combinatorial MatIt. V, 1-27 (lecture note un mathematices, 622. Springer-Verlag, Berlin, Heidelberg, New York, 1977). [6]Imrich, W., Qn finitely generated subgroups of free groups. ArcIt. MatIt. 28 (1977), 21.24. [7] Lyndon, R.C. and ScItupp, P.E., Combirtatonial group tircory. Engebniss vol. 89, Berlin Heidelberg New York: Springer (1977). -
-
[8] Neuniann, H., Qn tire intersection of finitely gevierated fra groups, MatIt. Debrecen 4 (1955-56), 186-189.
[9] Neumann, H., Qn tire intersection of finítely generated free groups: Addendum. MatIt Debrecen 5, (1957-50) 128.
[10]Nickolas, P., Intersection of finitely gerterated free groups. Austral. MatIt. Soc. Vol. 31(1985), 339-349
Buil.
[11] Magnus, W., Karrass, A. and Solltaar, D, Combinatorial group tireory. New York Wiley (1966). [12] Sarvatius, B., A siront proof of a tireonem of Burns. MatIt. Z. 184 (1983), 133-137. [13] Stallings, J.R., Topology of finite grapirs. Invent. MatIt. 71(1983),
551-565. PO.
fox 4286
Ádbamiyah
BAGHDAD IRAQ
Recibido: 6 de Junio, Revisado: 25 de Abril,
1994 1995
