On the Hanna Neumann Conjecture

Transcription

1 On the Hnn Neumnn Conjecture Toshiki Jitsukw Bill Khn Alexei G. Mysnikov Astrct The Hnn Neumnn conjecture sttes tht if F is free group, then for ll nontrivil finitely generted sugroups H, K F, rnk(h K) 1 [rnk(h) 1] [rnk(k) 1] Where most ppers to dte hve considered direct grph theoretic interprettion of the conjecture, here we consider the use of monomorphisms. We illustrte the effectiveness of this pproch with two results. First, we show tht for ny finitely generted groups H, K F either the pir H, K or the pir H, K stisfy the Hnn Neumnn conjecture; here denotes the utomorphism which sends ech genertor of F to its inverse. Next, using prticulr monomorphisms from F to F 2, we otin tht if the Hnn Neumnn conjecture is flse then there is counterexmple H, K F 2 hving the dditionl property tht ll the rnch vertices in the foldings of H nd K re of degree 3, nd ll degree 3 vertices hve the sme locl structure or type. 1 Introduction H. Neumnn proved in [12] tht ny nontrivil sugroups H, K f.g. F (finitely generted) must stisfy rnk(h K) 1 2[rnk(H) 1][rnk(K) 1], (1) nd so improved Howson s erlier result [5] tht H K is finitely generted. The stronger ssertion otined y omitting the fctor of 2 in (1) hs come to e known s the Hnn Neumnn conjecture. In [1], R. Burns improved H. Neumnn s ound y showing tht rnk(h K) 1 2[rnk(H) 1][rnk(K) 1] min(rnk(h) 1, rnk(k) 1). In 1983, J. Stllings introduced the notion of folding nd showed how to pply these ojects to the study of sugroups of free groups [16]. Stllings s pproch ws pplied y S. Gersten in [4] to solve certin specil cses of the conjecture, nd similr techniques were developed over sequence of ppers y W. Imrich [7, 6], P. Nickols [14], nd B. Servtius [15] who gve lternte 1

2 proofs of Burns ound nd resolved specil cses of the conjecture. In 1989, W. Neumnn showed tht the conjecture is true with proility 1 for rndomly chosen sugroups of free groups [13], nd proposed strengthened form of the conjecture which ounds the quntity r(h, K) def = r(h g K) g X where X F is set of doule coset representtives for the doule cosets HgK, nd r is the reduced rnk, defined s r(g) = mx(rnk(g) 1, 0) for sugroups G F. Specificlly, the strengthened Hnn Neumnn conjecture sserts: r(h, K) r(h) r(k) In 1992, G. Trdos proved in [17] tht the strengthened conjecture is true if one of the two sugroups hs rnk 2. Then, in 1994, W. Dicks showed tht the strengthened Hnn Neumnn conjecture is equivlent to conjecture on iprtite grphs, which he termed the Amlgmted Grph conjecture [2]. In 1996, G. Trdos used Dicks method to give the first new ound for the generl cse in [18], where he proved tht for ny H, K F with rnk(h), rnk(k) t lest 2, r(h, K) 2 r(h) r(k) r(h) r(k) + 1 Since then, W. Dicks nd E. Formnek [3] resolved the conjecture for the cse when one of the sugroups hs rnk t most 3, y proving tht r(h, K) r(h) r(k) + r 3 (H) r 3 (H), where r k (G) def = mx(rnk(g) k, 0) for sugroups G F nd k N. The strengthened conjecture ws lso recently solved in the specil cse when one of the two groups, sy H, hs generting set consisting of positive words (i.e. set of words in which no genertor of F hs negtive exponent). Specificlly, it ws shown y J. Mekin nd P. Weil [11], nd independently y B. Khn [9] tht if there is some utomorphism of F which crries generting set of H to set of positive words, then the conjecture holds for H nd ny nontrivil K f.g. F. Recll tht n utomorphism σ of F (X) is clled length-preserving if u F, u σ = u, i.e. (X ± ) σ = X ± where X ± = X X 1. In section 3, we shll prove the following two theorems: Theorem 1. Let F 2 = F (, ). Tke σ Aut(F 2 ) to e ny length-preserving utomorphism hving no non-trivil fixed points, nd let τ e ny monomorphism τ : w w, 2

3 where w, w F 2 re ritrry elements for which the words w nd w re reduced s written. Then for ll nontrivil H, K f.g. F 2, either the pir H, K or the pir H τσ, K stisfy the strengthened Hnn Neumnn conjecture. Theorem 2. Let F = F (X) nd e the utomorphism given y x x 1 (for ech x X). Then for ll nontrivil H, K f.g. F, either the pir H, K or the pir H, K stisfy the strengthened Hnn Neumnn conjecture. Note tht in the spcil cse when X = 2, Theorem 2 follows immeditely from Theorem 1 y considering τ to e the identity utomorphism, nd σ to e the utomorphism sending x x 1 for every x in X. Recll tht given H = w 1,, w n F, one my determine the ssocited Stllings folding Γ H = (V H, E H ), y the following constructive procedure (see [16]): Construct n directed cycles c 1 = (V 1, E 1 ),..., c n = (V n, E n ), where V i = w i. Then pick one vertex from ech of the cycles, nd identify this suset of vertices, denoting the resulting vertex s the identity vertex 1 H. Lel the edges of cycle c i y successive letters of w i, strting t vertex 1 H. Finlly, repetedly identify pirs of edges e, e for which lel(e) = lel(e ) [hed(e) = hed(e ) til(e) = til(e )]. Ech such identifiction is clled n edge-folding nd we sy tht the edge e (s well s e ) ws folded. Figure 1 illustrtes the process, which termintes in finitely mny steps yielding the folding Γ H. It is esy to verify tht the folding so otined is well-defined, nd moreover, is independent of the choice of generting set for H. It is not hrd to see tht the rnk of H is precisely E H V H + 1. w n w 1 u Γ H u 1 H folding process 1 H w 2 v w 4 v w 3 Figure 1: Contructing folding from rose. Now we consider sugroups of F 2 : If H f.g. F 2 then Γ H hs vertices of undirected degree 4, where y undirected degree d = d H (v) of vertex v we men the sum of the numer of outgoing nd incoming edges t v. Put d i (Γ H ) = {v V H d H (v) = i}, for i = 1, 2, 3, 4. Vertices of degree 3 my e 3

4 clssified into 4 types, denoted C, C, C 1, C 1, sed on the lels of the incident edges (see figure 2). For ech x { ±, ± }, we define C x (Γ H ) to e the numer of degree 3 vertices of type C x in Γ H. The rnk of H cn e computed y the formul rnk(h) = d 4 (Γ H ) + d 3(Γ H ) 2 d 1(Γ H ) C C C C 1 1 Figure 2: Locl structure of vertex lels in folding. The grph-theoretic pproch to the Hnn Neumnn conjecture is sed on the following key oservtions [16, 8]. Consider the product utomton Γ H Γ K, whose vertex set is V H V K nd two vertices (u 1, v 1 ) nd (u 2, v 2 ) re connected y n edge lelled x iff oth (u 1, v 1 ) E H nd (u 2, v 2 ) E K hve lel x. Given folding Γ, its core is defined to e the grph otined y repeted deletion of non-identity degree 1 vertices. We denote this grph s Γ. Let (1 H, 1 K ) ply the role of identity vertex in Γ H Γ K. Then it is not hrd to see tht the components of (Γ H Γ K ) re in one to one correspondence with {Γ H g K} where g vries over suitly chosen set of doule coset representtives for the doule cosets HgK. More specificlly, the connected component of (Γ H Γ K ) which contins (1 H, 1 K ), is isomorphic to Γ H K. Mny proofs of prtil results towrds the Hnn Neumnn conjecture require cse-y-cse nlysis sed on the numers nd types of degree 3 vertices present in the foldings of H, K. The next theorem hs implictions on the numer of cses which need to e considered in such rguments; it is proved in section 3. Theorem 3. Let F = F (X) e non-elin free group. There is monomorphism φ 0 : F F 2 into the free group of rnk 2 such tht: for ny groups H, K F, the foldings of H φ 0, K φ 0 f.g. F 2 hve the property tht ll their rnch vertices re of degree 3, nd ll degree 3 vertices hve the sme type. The previous theorem hs the following immedite corollry: Corollry 1. If the Hnn Neumnn conjecture is flse then there is counterexmple H, K f.g. F 2 hving the dditionl property tht ll the rnch vertices in the foldings of H nd K re of degree 3, nd ll degree 3 vertices hve the sme type. 4

5 2 Preliminries The numers C x (Γ H ) nd C x (Γ K ) llow one to compute upper ounds on the numers of vertices of degree 3 in Γ H Γ K nd hence in Γ H K. By considering suitle conjugte of H nd K we cn ssume d 1 (Γ H ) = d 1 (Γ K ) = 0. Remrk 1. It follows from the definition og the product folding tht d 4 (Γ H K ) d 4 (Γ H Γ K ) = d 4 (Γ H )d 4 (Γ K ), nd d 3 (Γ H K ) d 3 (Γ H Γ K ) = d 4 (Γ H )d 3 (Γ K ) + d 3 (Γ H )d 4 (Γ K ) + x {,} ± C x (Γ H )C x (Γ K ) Definition 1. Given two sugroups H, K f.g. F 2 nd x {, } ±, we define { Cx (Γ H ) δ x (H, K) = min d 3 (Γ H ), C } x(γ K ) d 3 (Γ K ) δ(h, K) = mx δ x (H, K) x { ±, ± } nd put µ(h, K) to e ny x { ±, ± } for which δ x (H, K) = δ(h, K). Remrk 2. Wlter Neumnn [13] showed tht if H, K f.g. F 2 re counterexmple to the strengthened conjecture, then δ(h, K) > 1 2. We outline his rgument here, in grph-theoretic nottion. Using simple nd eutiful rgument from convexity theory, he showed tht if δ(h, K) 1 2 then C x (Γ H )C x (Γ K ) 1 2 d 3(Γ H )d 3 (Γ K ). x {,} ± Since the components of (Γ H Γ K ) re in one to one correspondence with {Γ H g K} (where g vries over suitly chosen set of doule coset representtives for the doule cosets HgK), we see tht r(h, K) = d 4 ((Γ H Γ K ) ) + d 3 ((Γ H Γ K ) )/2 d 1 ((Γ H Γ K ) )/2 which is t most d 4 (Γ H Γ K ) + d 3 (Γ H Γ K )/2. By Remrk 1 then, r(h, K) d 4 (Γ H Γ K ) + d 3 (Γ H Γ K )/2 d 4 (Γ H )d 4 (Γ K ) + 1 d 4 (Γ H )d 3 (Γ K ) + d 3 (Γ H )d 4 (Γ K ) 2 + C x (Γ H )C x (Γ K ) x {,} ± d 4 (Γ H )d 4 (Γ K ) + d 4 (Γ H )d 3 (Γ K )/2 + d 3 (Γ H )d 4 (Γ K )/2 r(h) r(k). nd thus the strengthened conjecture holds. + d 3 (Γ H )d 3 (Γ K )/4 5

6 3 Results Remrk 3. Given n endomorphism φ : F 2 F 2, the folding Γ H φ cn e otined from Γ H s follows. First construct lelled directed grph φ(γ H ) y replcing ech edge with lel x in Γ H y sequence of edges lelled y the successive letters of x φ (for x =, ). Then, pply the previously descried folding procedure to trnsform the grph φ(γ H ) Γ H φ. One my verify tht this yields folding which is isomorphic to the one otined y constructing Γ H φ directly from the set {w φ 1,, wφ n}. For exmple, if φ is length-preserving utomorphism, then Γ H φ cn e otined from Γ H y replcing every lel x y x φ nd chnging the orienttion of the edges if necessry. Lemm 1. Let Γ H e the folding of sugroup H f.g. F 2, nd φ : F 2 F 2 n endomorphism. If two edges e, f from φ(γ H ) get folded during the folding process φ(γ H ) Γ H φ, then there must exist pth p in φ(γ H ) eginning t e nd ending t f with the property tht every edge in p ws folded during the folding process. Proof. The lemm is proved y induction on the numer n of edge-foldings which tke plce during the folding process note tht this numer does not depend on the folding process since it is equl to E φ(γh ) E ΓH nd the φ resultnt folded grph Γ H φ is unique). For n = 1, the pth p consists of just edges e, f. Now suppose the first edge-folding occurs when edges d 1 nd d 2 re merged into n edge d, nd denote the folding otined fter this identifiction s Γ. By induction, there exists pth p in Γ connecting e nd f. There re two cses to consider: either d ppers in p, or it does not. In the first cse, let p 1 (resp. p 2 ) e the pth otined y replcing d with d 1 (resp d 2 ) in p. It is cler tht either p 1 or p 2 must fulfill the requirements of the lemm. In the second cse, we simply tke p = p. Lemm 2. Let H, K f.g. F 2 e sugroups which stisfy δ(h, K) > 1 2 (nd hence re potentil counterexmple to the strengthened Hnn Neumnn conjecture) nd tke, to e two length-preserving utomorphisms of F 2 whose vlues differ on µ(h, K), i.e. µ(h, K) µ(h, K). Then the groups H, K must stisfy the strengthened Hnn Neumnn conjecture. Proof. Set x 0 = µ(h, K). Since δ(h, K) > 1 2, it follows tht C x0 (Γ H ) > 1 2 d 3(Γ H ) C x0 (Γ K ) > 1 2 d 3(Γ K ). In light of Remrk 3, Γ K is the sme grph s Γ K, except tht ll edges hve een relelled s, nd edges hve een relelled s, nd n 6

7 nlogous sttement is true out the reltionship etween Γ H nd Γ H. So for ll x { ±, ± }. It follows tht C x (Γ K ) = C x (Γ K ) C x (Γ H ) = C x (Γ H ) C x 0 (Γ K ) = C x0 (Γ K ) > 1 2 d 3(Γ K ) = 1 2 d 3(Γ K ), (2) C x 0 (Γ H ) = C x0 (Γ H ) > 1 2 d 3(Γ H ) = 1 2 d 3(Γ H ). (3) Since x 0 x 0, it follows tht δ x (H, K ) < 1 2 for every x {, }±. Thus, δ(h, K ) < 1 2, nd hence y Remrk 2 the groups H, K cnnot e counterexmple to the strengthened Hnn Neumnn conjecture. The proofs of Theorems 1 nd 2 (see pge 2) now follow from Lemm 2. Proof. (Theorem 1) Suppose H, K do not stisfy the strengthened conjecture. By remrk 2 we hve δ(h, K) > 1 2. By definition of τ we hve δ(hτ, K) = δ(h, K). We pply Lemm 2 to H τ, K, tking to e the fixed-point-free length-preserving utomorphism σ, nd to e the identity utomorphism. The theorem follows. Proof. (Theorem 2) Suppose X = { 1,..., n }. We consider the emedding ψ : F (X) F 2 = F ( 1, 2 ) defined y ψ : i i 1 2 i 1. If H, K f.g. F (X) re counterexmple to the strengthened conjecture, then so re H ψ, K ψ f.g. F 2. Let e the utomorphism of F (X) given y i 1 i (for ech i X). Restricting to F 2 nd pplying the previous lemm, we see tht either H ψ, K ψ or (H ψ ), K ψ must stisfy the strengthened conjecture. But (H ) ψ = (H ψ ) ; here we think of H ψ s sugroup of F (X) under the cnonicl inclusion of F 2 into F (X). It follows tht either H ψ, K ψ or (H ) ψ, K ψ must stisfy the strengthened conjecture. Since ψ is monomorphism, this implies tht either H, K or H, K must stisfy the strengthened conjecture. Now towrds the proof of Theorem 3, we introduce the following definition: Definition 2. We sy φ : F 2 (, ) F 2 (, ) is N-endomorphism if it hs the property tht U φ = { φ, φ } is N-reduced [10, pp.6], which is to sy tht every triple v 1, v 2, v 3 in U ± φ stisfies (N0) v 1 1, (N1) v 1 v 2 1 implies v 1 v 2 v 1, v 2, (N2) v 1 v 2, v 2 v 3 1 implies v 1 v 2 v 3 > v 1 v 2 + v 3. 7

8 Remrk 4. It is well-known [10, pp.7] tht if suset U of free group F stisfies N0-N2, then one my ssocite with ech u U words (u), m(u) F with m(u) 1 such tht u = (u)m(u)(u 1 ) 1 in F nd hving the property tht for ny w = u 1 u t, t 0, u i U ± where u i u i+1 1, the suwords m(u 1 ),..., m(u t ) remin uncncelled in the reduced form of w. Lemm 3. Every N-endomorphism of F 2 is monomorphism. Proof. Tke w F 2, with w 1. By (N2), (w φ ) 3 > 0, hence (w φ ) 3 1. It follows tht w φ 1. Lemm 4. Given H f.g. F 2 nd n N-endomorphism φ of F 2, then for every edge e in Γ H, t lest one edge from the imge of e under φ does not get folded during the folding process φ(γ H ) Γ H φ. Proof. Let e = (u, v) e ny edge of Γ H ; suppose e is lelled y x {, } ±. Consider the pth φ(e) in φ(γ H ); this pth consists of sequence of edges lelled y successive letters of x φ. Since φ is n N-endomorphism, { φ, φ } is N-reduced nd Remrk 4 pplies. Accordingly, let ē e the edge in φ(e) which corresponds to the first letter of m(x φ ) inside x φ. We clim tht ē does not get folded during the folding process φ(γ H ) Γ H φ. Suppose towrds contrdiction, tht ē gets folded with some edge f during the folding process φ(γ H ) Γ H φ. Then y Lemm 1, there must exist noncktrcking pth p in φ(γ H ) eginning t ē nd ending t f, with the property tht every edge in p ws folded during the folding process. Since p is noncktrcking pth in φ(γ H ), it is supth of φ(q) for some non-cktrcking pth q in Γ H. It follows tht the lels long φ(q) re word u 1 u t, t 0, u i { φ, φ } ± nd u i u i+1 1. Since ē is lelled y the first letter of m(x φ ), y Remrk 4 the edge ē ws not folded during the folding process; this is contrdiction. We introduce the following nottions: let Γ H = (V H, E H ) e the folding of H. Tke ny vertex v V H, nd let E v e the edges incident to v. Define Γ v to e the tree sugrph of Γ H induced y edges E v. Then φ(γ v ) is lso tree. By Lemm 4, we my ssocite to ech edge e E v, n edge m(e) E φ(γh ) which does not get folded during the folding process φ(γ H ) Γ H φ. We define trφ(γ v ) to e the grph otined y truncting the rnches of φ(γ v ) so tht they terminte with edges m(e), e E v. Clerly, for ll v V H, trφ(γ v ) is sugrph of φ(γ H ). The next lemm shows tht N-endomorphisms do not cuse lrge-scle disturnces in the neighorhood of rnch vertices. Lemm 5. Given H f.g. F 2 nd n N-endomorphism φ of F 2, then during the folding process φ(γ H ) Γ H φ, no edge from trφ(γ v ) gets folded with n edge from outside trφ(γ v ). Proof. Suppose, towrds contrdiction, tht n edge e inside trφ(γ v ) nd n edge f outside trφ(γ v ) get folded during the folding process φ(γ H ) Γ H φ. 8

9 Then y Lemm 1, there exists pth p eginning t e nd ending t f with the property tht every edge in p ws folded during the folding process. Then p must pss through some edge m(e), e E v. This contrdicts the properties of m(e) s determined in Lemm 4. It follows tht no edge inside trφ(γ v ) gets folded with n edge outside trφ(γ v ). Informlly stted, the previous lemm implies tht for n N-endomorphism φ nd sugroup H f.g. F 2, the 5-tuple of vlues C (Γ H φ), C (Γ H φ), C 1(Γ H φ), C 1(Γ H φ), d 4 (Γ H φ) is completely determinle from the 5-tuple of vlues C (Γ H ), C (Γ H ), C 1(Γ H ), C 1(Γ H ), d 4 (Γ H ), without knowledge of ny further structure (e.g. the generting set) of H. Lemm 6. Let φ 0 : F 2 F 2 e the endomorphism defined y φ 0 = 2 nd φ 0 = [, ]. Then for ny finitely generted sugroup H f.g. F 2, C 1(Γ H φ 0 ) = d 3 (Γ H φ 0 ), nd rnk(h φ0 ) = rnk(h). Proof. It is strightforwrd to check tht { 2, [, ]} is N-reduced, nd hence φ 0 is n N-endomorphism. By Lemm 5 it suffices to consider the effect of φ 0 on the vrious types of rnch vertices in Γ H. Figure 3 depicts how φ 0 trnsforms φ 0 Figure 3: The effect of φ 0 on vertex of degree 4. the neighorhood of degree 4 vertex v to produce two vertices of type C 1. The effect of φ 0 on ech of the four types of degree 3 vertices my e determined simply y restricting our considertion to pproprite sugrphs of the depicted neighorhood. It follows tht φ 0 trnsforms ny degree 3 vertex in Γ H into vertex of type C 1 in Γ H φ 0. Thus, C 1(Γ H φ 0 ) = d 3 (Γ H ) + 2d 4 (Γ H ). Since ll rnch vertices of Γ H re seen to produce vertices of type C 1, we get tht C 1(Γ H φ 0 ) = d 3 (Γ H φ 0 ). Finlly, y Lemm 3, φ 0 is monomorphism, so rnk(h φ 0 ) = rnk(h). The proof of Theorem 3 (see pge 4) now follows from Lemm 6. 9

10 Proof. (Theorem 3) Fix the emedding ψ : F (X) F 2 = F (, ) defined y ψ : i i i. Put H = (H ψ ) φ 0 nd K = (K ψ ) φ 0, where φ 0 is the endomorphism of F (, ) defined y φ0 = 2 nd φ0 = [, ]. Then y Lemm 6, nd hence δ(h, K ) = 1. C 1(Γ H ) = d 3 (Γ H ) C 1(Γ K ) = d 3 (Γ K ), Remrk 5. The following somewht simpler proof for Theorem 3 ws communicted to us y W. Neumnn: Consider n N-endomorphism φ 1 : F 2 F 2 with the dditionl property tht the grph for φ 1 (F 2 ) hs only type vertices (e.g. 1 nd 1 1. If H is ny sugroup of F 2 then the grph for φ 1 (H) is sugrph of covering grph of the grph for φ 1 (F 2 ), so it lso hs only type vertices, hence H = (H ψ ) φ 1 nd K = (K ψ ) φ 1 stisfy δ(h, K ) = 1 for ll H, K F (X). Corollry 1 follows immeditely from Theorem 3, since φ 0, φ 1 nd ψ re monomorphisms nd hence rnk(h) = rnk(h ), rnk(k) = rnk(k ). 4 Acknowledgements We re grteful to thnk W. Neumnn nd A. Mgidin for their insightful comments nd suggestions. The first two uthors re grteful to the Mthemtics deprtment t the City University of New York Grdute Center for funding this work s prt of the their ongoing doctorl reserch. The second uthor thnks oth the Center for Computtionl Sciences t the Nvl Reserch Lortory in Wshington DC, nd the Advnced Engineering nd Sciences division of ITT Industries for their support of these endevors. References [1] R. G. Burns. On the intersection of finitely generted sugroups of free group. Mthe. Z., 119: , [2] Wrren Dicks. Equivlence of the strengthened Hnn Neumnn conjecture nd the mlgmted grph conjecture. Invent. Mth., 117: , [3] Edwrd Formnek nd Wrren Dicks. The rnk three cse of the Hnn Neumnn conjecture. J. Group Theory, 4: , [4] S. M. Gersten. Intersections of finitely generted sugroups of free groups nd resolutions of grphs. Invent. Mth., 71: ,

11 [5] A. G. Howson. On the intersection of finitely generted free groups. J. London Mth. Soc., 29: , [6] W. Imrich. On finitely generted sugroups of free groups. Arch. Mth., 28:21 24, [7] W. Imrich. Sugroup theorems nd grphs. Comintoril Mthemtics V, Lecture notes in mthemtics, vol. 622:1 27, Melourne, [8] Ily Kpovich nd Alexei Misnikov. Stllings foldings nd the sugroup structure of free groups. J. Alger, 248 no 2: , [9] Bill Khn. Positively generted sugroups of free groups nd the Hnn Neumnn conjecture. In Contemporry Mthemtics, volume 296. R. Gilmn et l. Americn Mthemticl Society, [10] Roger C. Lyndon nd Pul E. Schupp. Comintoril Group Theory. Springer-Verlg, New York, [11] John Mekin nd Pscl Weil. Sugroups of free groups: contriution to the Hnn Neumnn conjecture. Geom. Dedict, to pper. [12] H. Neumnn. On intersections of finitely generted sugroups of free groups. Pul. Mth., Derecen, 5: , [13] W. Neumnn. On the intersection of finitely generted free groups. In Cnerr 1989 (Lecture notes in Mth., vol. 1456, pp ). Springer- Verlg, [14] P. Nickols. Intersections of finitely generted free groups. Bull. of the Austrlin Mth. Soc., 34: , [15] B. Servtius. A short proof of theorem of Burns. Mthe. Z., 184: , [16] J. R. Stllings. Topology of finite grphs. Invent. Mth., 71: , [17] G. Trdos. On the intersection of sugroups of free group. Invent. Mth., 108:29 36, [18] G. Trdos. Towrd the Hnn Neumnn Conjecture using Dicks method. Invent. Mth., 123:95 104,