1 Introduction
Recall that the Hanna Neumann conjecture [16] claims that if F is a free group of rank
r(F), r¯(F):=max(r(F)-1,0) is the reduced rank of F, and H1, H2
are finitely generated subgroups of F, then r¯(H1∩H2)≤r¯(H1)r¯(H2). It was shown by Hanna
Neumann [16] that r¯(H1∩H2)≤2r¯(H1)r¯(H2). For more discussion and results on this problem, the reader is referred to
[2, 3, 6, 15, 17, 18].
More generally, let ℱ=∏α∈I*Gα be the free product of some groups
Gα, α∈I. According to the classic Kurosh subgroup
theorem [13], every subgroup H of ℱ
is a free product F(H)*∏*sα,βHα,βsα,β-1, where Hα,β is a subgroup of Gα, sα,β∈ℱ, and F(H) is a free subgroup of
ℱ such that, for every s∈ℱ
and γ∈I, it is true that F(H)∩sGγs-1={1}. We say that H
is a factor-free subgroup of ℱ if H=F(H) in the above form of H, i.e., for every s∈ℱ and γ∈I, we have H∩sGγs-1={1}. Since a factor-free subgroup H of
ℱ is free, the reduced
rank r¯(H):=max(r(H)-1,0) of H is well defined.
Let q*=q*(Gα,α∈I) denote the minimum of orders >2 of finite subgroups of groups Gα, α∈I, and q*:=∞ if there are no such subgroups. If q*=∞, define
q*q*-2:=1. It was shown by Dicks and the author [4] that if
H1 and H2 are finitely generated factor-free subgroups of ℱ, then
r¯(H1∩H2)≤2q*q*-2r¯(H1)r¯(H2).
It was conjectured by Dicks and the author [4] that if groups Gα, α∈I, contain no involutions, then – similarly to the
Hanna Neumann conjecture – the coefficient 2 could be left out and
(1.1)r¯(H1∩H2)≤q*q*-2r¯(H1)r¯(H2).
A special case of this generalization of the Hanna Neumann conjecture is established by Dicks and the author [5] by proving that if H1 and H2 are finitely generated factor-free subgroups of the free product
ℱ all of whose factors are groups of order 3, then, indeed,
r¯(H1∩H2)≤q*q*-2r¯(H1)r¯(H2)=3r¯(H1)r¯(H2).
We remark that it follows from results of [4] that the last inequality (as well as (1.1)) is sharp and may not be improved.
Here is another special case when the conjectured inequality (1.1) holds true.
Theorem 1.
Let Gα, α∈I, be left (or right) ordered groups, let
F=∏α∈I*Gα be their free product, and let
H1 and H2 be finitely generated factor-free subgroups of F. Then
r¯(H1∩H2)≤q*q*-2r¯(H1)r¯(H2)=r¯(H1)r¯(H2).
Moreover, let S(H1,H2) denote a set of representatives of those double cosets H1tH2 of F, t∈F, that have the property H1∩tH2t-1≠{1}. Then
r¯(H1,H2):=∑s∈S(H1,H2)r¯(H1∩sH2s-1)≤r¯(H1)r¯(H2).
We remark that Antolín, Martino and Schwabrow [1] proved a more general result on Kurosh rank of the intersection of subgroups of free products of right ordered groups by utilizing the Bass–Serre theory of groups acting on trees and some ideas of Dicks [3]. Our proof of Theorem 1 seems to be of independent interest as it uses explicit geometric construction of graphs, representing subgroups of free products, that are often more suitable for counting arguments, see [9, 11].
It is fairly easy to see that Theorem 1 implies both the Hanna Neumann conjecture and the strengthened Hanna Neumann conjecture, put forward by W. Neumann [17], see Section 5. The strengthened Hanna Neumann conjecture claims that
if H1 and H2 are finitely generated subgroups of a free group F, then
∑s∈S(H1,H2)r¯(H1∩sH2s-1)≤r¯(H1)r¯(H2),
where the set S(H1,H2) is defined as in Theorem 1 with F in place in ℱ.
Recall that Friedman [6] proved the strengthened Hanna Neumann conjecture by making use of sheaves on graphs and Mineyev [15] gave a proof of the strengthened Hanna Neumann conjecture by using Hilbert modules and group actions, see also Dicks’s proof [3]. Similarly to Dicks [3] and Mineyev [15], we also use the idea of group ordering and special sets of edges, however, our arguments deal directly with core graphs of subgroups of free products that are analogous to Stallings graphs [18] representing subgroups of free groups.
In Section 2, we introduce the necessary definitions and basic terminology. In Section 3, we define and study strongly positive words in free products of left ordered groups. Section 4 contains the proof of Theorem 1. In Section 5, we briefly look at the case of free groups.
2 Preliminaries
Let Gα, α∈I, be nontrivial
groups, ℱ=∏α∈I*Gα their free product, and H a finitely generated factor-free subgroup of ℱ, H≠{1}.
Consider an alphabet 𝒜=⋃α∈IGα, where Gα∩Gα′={1} if
α≠α′.
Like the graph-theoretic approach of article [9], in a simplified version suitable for finitely generated factor-free subgroups of ℱ, see [7, 8, 10], we first define a labeled 𝒜-graph Ψ(H) which geometrically represents H in a fashion analogous to the way the Stallings graph represents a subgroup of a free group, see [18].
If Γ is a graph, VΓ denotes the vertex set of Γ and EΓ denotes the set of oriented edges of Γ. If e∈EΓ, e-, e+ denote the initial, terminal, respectively, vertices of e and e-1 is the edge with opposite orientation, where e-1≠e for every e∈EΓ, (e-1)-=e+, (e-1)+=e-.
A finite pathp=e1…ek in Γ is a sequence of edges ei such that (ei)+=(ei+1)-, i=1,…,k-1.
Denote p-:=(e1)-, p+:=(ek)+, |p|:=k, where |p| is the length of p. We allow
the possibility |p|=0, p={p-}={p+}. A finite path p is called closed if p-=p+. An infinite path p=e1…ek… is an infinite sequence of edges ei such that (ei)+=(ei+1)- for i=1,2,…. If p=e1…ek… and q=f1…fℓ… are infinite paths such that (e1)-=(f1)-, then
q-1p:=…fℓ-1…f1-1e1…ek… is a biinfinite path.
A path p is reduced if p contains no subpath of the form ee-1, e∈EΓ.
A closed path p=e1…ek is cyclically reduced if |p|>0 and both p and the cyclic permutation
e2…eke1 of p are reduced paths. The core of a graph Γ, denoted core(Γ),
is the minimal subgraph of Γ that contains every edge e which can be included into a cyclically reduced path in Γ.
Let Ψ be a graph whose vertex set VΨ consists of two disjoint parts VPΨ and VSΨ, so VΨ=VPΨ∪VSΨ. Vertices in VPΨ are called primary and vertices in VSΨ are termed secondary.
Every edge e∈EΨ connects primary and secondary vertices, hence Ψ is a bipartite graph; Ψ is called a labeled A-graph, or simply 𝒜-graph if
Ψ is equipped with a map φ:EΨ→𝒜, called a labeling, such that, for every e∈EΨ, φ(e)∈𝒜=⋃α∈IGα, φ(e-1)=φ(e)-1 and if
e+=f+∈VSΨ, then φ(e),φ(f)∈Gα for the same α=θ(e+)∈I, called the type of the vertex e+∈VSΨ and denoted α=θ(e+). If e+∈VSΨ, define θ(e):=θ(e+) and θ(e-1):=θ(e+). Thus, for every e∈Ψ, we have defined an element
φ(e)∈𝒜, called the label of e, and θ(e)∈I, the type of e.
The reader familiar with van Kampen diagrams over a free product of groups,
see [14], will recognize that our labeling function φ:EΨ→𝒜 is defined in a way analogous to labeling functions on such diagrams. Recall that van Kampen diagrams are planar 2-complexes whereas graphs are 1-complexes, however, apart from this, the ideas of cancelations and edge foldings work equally well for both diagrams and graphs.
An 𝒜-graph Ψ is called irreducible if properties (P1)–(P3) hold true:
If e,f∈EΨ, e-=f-∈VPΨ, and e+≠f+, then θ(e)≠θ(f).
If e,f∈EΨ, e≠f, and e+=f+∈VSΨ, then φ(e)≠φ(f) in Gθ(e).
Ψ has no multiple edges, degΨv>0 for every v∈VΨ, and there is at most one vertex of degree 1 in Ψ which, if it exists, is primary.
Suppose Ψ is a connected
irreducible 𝒜-graph and a primary vertex o∈VPΨ is distinguished so that degoΨ=1 if Ψ happens to have a vertex of degree 1. Then o is called the base vertex of Ψ=Ψo.
As usual, elements of the free product ℱ=∏α∈I*Gα
are regarded as words over the alphabet 𝒜=⋃α∈IGα. A
syllable of a word W over 𝒜 is a maximal subword of W
all of whose letters belong to the same factor Gα. The syllable length∥W∥ of W is the number of syllables
of W, whereas the length|W| of W is the number of
letters in W.
A nonempty word W over 𝒜 is called reduced if every
syllable of W consists of a single letter.
Clearly, |W|=∥W∥ if W is reduced. An arbitrary nontrivial element of the
free product ℱ can be uniquely
written as a reduced word. A word W is called cyclically reduced if W2 is reduced.
We write U=W if words U and W
are equal as elements of ℱ. The
literal (or letter-by-letter) equality of words U and W is denoted U≡W.
The significance of irreducible 𝒜-graphs for geometric interpretation of factor-free subgroups
H of ℱ is given in the following.
Lemma 1.
Suppose H is a finitely generated factor-free subgroup
of the free product F=∏α∈I*Gα, H≠{1}. Then there exists a finite connected
irreducible A-graph Ψ=Ψo(H), with base vertex o, such that a reduced word W over the alphabet A belongs to H if and only if there is a reduced path p in Ψo(H) such that p-=p+=o, φ(p)=W in F, and |p|=2|W|.
Proof.
The proof is based on Stallings’s folding techniques and is somewhat analogous to the proof of the van Kampen lemma for diagrams over free products of groups, see [14] (in fact, it is simpler because foldings need not preserve the property of being planar for the diagram). A more general approach, suitable for an arbitrary subgroup of ℱ, is discussed in [9, Lemmas 1 and 4].
Let H=〈V1,…,Vk〉 be generated by reduced words V1,…,Vk∈ℱ. Consider a graph Ψ~ which consists of k closed paths p1,…,pk such that they have a single common vertex o=(pi)-, and |pi|=2|Vi|, i=1,…,k.
Furthermore, we distinguish o as the base vertex of Ψ~ and call o primary, the vertices adjacent to o are called secondary vertices and so on. The labeling function φ on pi is defined so that φ(pi)=Vi, i=1,…,k, where φ(p):=φ(e1)…φ(eℓ) if p=e1…eℓ and e1,…,eℓ∈EΨ~.
Clearly, Ψ~=Ψ~o is a finite connected 𝒜-graph with the base vertex o that has the following property
A word W∈ℱ belongs to H if and only if there is a
path p in Ψ~o such that p-=p+=o and φ(p)=W.
However, Ψ~o need not be irreducible and we will do foldings of edges in Ψ~o which preserve property (Q) and which aim to achieve properties (P1)–(P2).
Assume that property (P1) fails for edges e,f with e-=f-∈VPΨ~o so that e+≠f+ and θ(e)=θ(f).
Let us redefine the labels of all edges e′ with e+′=e+ so that φ(e′)φ(e)-1 does not change
and φ(e)=φ(f) in Gθ(e). Now we identify the edges e, f and vertices e+, f+. Observe that this folding preserves property (Q) ((P2) might fail) and decreases the edge number |EΨ~o|.
If property (P2) fails for edges e,f and φ(e)=φ(f) in Gθ(e), then we identify the edges e,f. Note property (Q) still holds ((P1) might fail) and the number |EΨ~o| decreases.
Suppose property (P3) fails and there are two distinct edges e,f in Ψ~o such that e-=f-, e+=f+∈VSΨ~o. By property (Q), it follows from H being factor-free that φ(e)=φ(f) in Gθ(e).
Therefore, we can identify the edges e,f, thus preserving property (Q) and decreasing the number |EΨ~o|. If property (P3) fails so that there is a vertex v of degree 1, different from o, then we delete v along with the incident edge. Clearly, property (Q)
still holds and the number |EΨ~o| decreases.
Thus, by induction on |EΨ~o|, in polynomial time of size of input, which is the total length ∑i=1k|Vi|, we can effectively construct an irreducible 𝒜-graph Ψo with property (Q). Other stated properties of Ψo are straightforward.
∎
The following lemma further elaborates on the correspondence between finitely generated
factor-free subgroups of the free product ℱ and irreducible 𝒜-graphs.
Lemma 2.
Let Ψo be a finite connected
irreducible A-graph with the base vertex o,
and let H=H(Ψo) be a subgroup of
F that consists of all words φ(p), where p is
a path in Ψo with p-=p+=o. Then H is a
factor-free subgroup of F and
r¯(H)=-χ(Ψo), where χ(Ψo)=|VΨo|-12|EΨo| is
the Euler characteristic of Ψo.
Proof.
This follows from the facts that the fundamental group π1(Ψo,o) of
Ψo at o is free of rank -χ(Ψo)+1 and that the homomorphism π1(Ψo,o)→ℱ, given by p→φ(p), where p is a path with p-=p+=o, has the trivial kernel following from properties (P1)–(P2).
∎
Suppose H is a nontrivial finitely generated factor-free subgroup of a free product ℱ=∏α∈I*Gα, and Ψo=Ψo(H) is an irreducible 𝒜-graph for H as in Lemma 1. Let Ψ(H) denote the core of Ψo(H). Clearly, Ψ(H)
has no vertices of degree ≤1 and Ψ(H) is also an irreducible 𝒜-graph. It is easy to see that the graph Ψo(H) of H can be obtained back from the core graph Ψ(H) of H by attaching a suitable path p to Ψ(H) so that p starts at a primary vertex o, ends in p-∈VPΨ(H), and then by doing foldings of edges as in the proof of Lemma 1.
Now suppose H1 and H2 are nontrivial finitely generated factor-free subgroups of ℱ.
Consider a set S(H1,H2) of representatives of those double cosets
H1tH2 of ℱ, t∈ℱ, that have the property H1∩tH2t-1≠{1}.
For every s∈S(H1,H2), define the subgroup Ks:=H1∩sH2s-1.
Analogously to the case of free groups, see [17, 18], we now
construct a finite irreducible 𝒜-graph Ψ(H1,H2) whose connected components are core graphs Ψ(Ks), s∈S(H1,H2).
First we define an 𝒜-graph Ψo′(H1,H2). The set of primary vertices of the graph
Ψo′(H1,H2) is
VPΨo′(H1,H2):=VPΨo1(H1)×VPΨo2(H2).
Let
τi:VPΨo′(H1,H2)→VPΨoi(Hi)
denote the projection map, τi((v1,v2))=vi, i=1,2.
The set of secondary vertices VSΨo′(H1,H2) of
Ψo′(H1,H2) consists of equivalence classes [u]α, where u∈VPΨo′(H1,H2), α∈I, with respect to the minimal equivalence relation generated by the following relation ∼𝛼 on VPΨo′(H1,H2): Define v∼𝛼w if there are edges ei,fi∈EΨoi(Hi) such that
(ei)-=τi(v),(fi)-=τi(w),(ei)+=(fi)+,i=1,2,
the edges ei,fi have type α, and
φ(e1)φ(f1)-1=φ(e2)φ(f2)-1
in Gα. It is easy to see that ∼𝛼 is symmetric and transitive on distinct pairs and triples (but it could lack the reflexive property).
The edges in the 𝒜-graph Ψo′(H1,H2) are defined so that u∈VPΨo′(H1,H2) and
[v]α∈VSΨo′(H1,H2) are connected by an edge if and only if u∈[v]α.
The type of a vertex [v]α∈VSΨo′(H1,H2) is α and if e∈EΨo′(H1,H2), e-=u and e+=[v]α, then φ(e):=φ(e1), where e1∈EΨo1(H1)
is an edge of type α with (e1)-=τ1(u), when such an e1 exists, and φ(e1):=bα≠1, bα∈Gα, otherwise.
It follows from the definitions and properties (P1)–(P2) of Ψoi(Hi), i=1,2, that Ψo′(H1,H2) is an 𝒜-graph with properties (P1)–(P2). Hence, taking the core of
Ψo′(H1,H2), we obtain an irreducible 𝒜-graph which we denote Ψ(H1,H2).
In addition, it is not difficult to see that, when taking the connected component Ψo′(H1,H2,o) of Ψo′(H1,H2) that contains the vertex o=(o1,o2) and inductively removing from Ψo′(H1∩H2,o) vertices of degree 1 different from o, we obtain an irreducible 𝒜-graph Ψo(H1∩H2) with base vertex o that corresponds to the intersection H1∩H2 as in Lemma 1.
Observe that it follows from the definitions and property (P1) for Ψ(Hi), i=1,2, that, for every edge e∈EΨ(H1,H2) with e-∈VPΨ(H1,H2), there are unique edges
ei∈EΨ(Hi) such that τi(e-)=(ei)-, i=1,2. Hence, by setting τi(e)=ei, τi(e+)=(ei)+, i=1,2, we extend τi to the graph map
τi:Ψ(H1,H2)→Ψ(Hi),i=1,2.
It follows from the definitions that τi is locally injective and τi preserves syllables of φ(p) for every path p with primary vertices p-,p+.
Lemma 3.
Suppose H1 and H2 are finitely generated factor-free
subgroups of the free product F and the set
S(H1,H2) is not empty. Then the connected components of the graph Ψ(H1,H2) are core graphs Ψ(H1∩sH2s-1) of subgroups H1∩sH2s-1, s∈S(H1,H2). In particular,
r¯(H1,H2)=∑s∈S(H1,H2)r¯(H1∩sH2s-1)=-χ(Ψ(H1,H2)).
Proof.
As in Lemma 1, let Ψoi(Hi) be an irreducible 𝒜-graph, corresponding to the subgroup Hi of ℱ, i=1,2, and Ψ(Hi) denote the core of Ψoi(Hi).
Let vi∈VPΨ(Hi), i=1,2, and q(vi) denote a path in
Ψoi(Hi) with q(vi)-=oi and q(vi)+=vi. Suppose Xi∈ℱ for i=1,2, and H1X1∩H2X2≠{1}, where HiXi:=XiHiXi-1. Consider
an irreducible 𝒜-graph Ψui(HiXi), i=1,2. Note that the core graph
Ψ(H1X1∩H2X2) can be identified with a connected component, denoted Ψ(X1,X2)(H1,H2),
of the irreducible 𝒜-graph Ψ(H1,H2). In addition, if w∈VPΨ(X1,X2)(H1,H2), there are paths pi(w) in Ψui(HiXi), i=1,2, such that (pi(w))-=ui, (pi(w))+=τi(w), and φ(p1(w))=φ(p2(w)). Furthermore, it follows from the definitions that Xiφ(q(τi(w)))φ(pi(w))-1∈HiXi, i=1,2.
Therefore, there are words Vi∈Hi, i=1,2, such that
XiViφ(q(τi(w)))=φ(pi(w)).
Since φ(p1(w))=φ(p2(w)), we further obtain
(2.1)X1-1X2=V1φ(q(τ1(w)))φ(q(τ2(w)))-1V2-1.
Now we can draw the following conclusion. For every pair (X1,X2)∈ℱ×ℱ such that H1X1∩H2X2≠{1} and vertex
w∈VPΨ(X1,X2)(H1,H2), there are words Vi∈Hi, i=1,2, such that equality (2.1) holds. As the paths q(τi(w)), i=1,2, in (2.1) depend only on a connected component of Ψ(H1,H2), it follows from (2.1) that if
(1,X),(1,Y) are some pairs such that Ψ(1,X)(H1,H2)=Ψ(1,Y)(H1,H2), then X∈H1YH2⊆ℱ.
Conversely, if X∈H1YH2, the equality Ψ(1,X)(H1,H2)=Ψ(1,Y)(H1,H2)
is obviously true. Thus, the set S(H1,H2) is in bijective correspondence with connected components of Ψ(H1,H2) and, by Lemma 2, we have
r¯(H1∩sH2s-1)=-χ(Ψ(1,s)(H1,H2))
for every s∈S(H1,H2). Adding up over all s∈S(H1,H2), we arrive at the required equality r¯(H1,H2)=-χ(Ψ(H1,H2)).
∎
3 Strongly positive words in free products of left ordered groups
Recall that G is called a left ordered group
if G is equipped with a total order ≤ which is left invariant, i.e.,
for every triple a,b,c∈G, the relation a≤b implies ca≤cb. If G is left ordered, then G can also be right ordered (and vice versa). Indeed, if ≤ is a left order on G then, setting a⪯b if and only if a-1≤b-1, we obtain a right order ⪯ on G.
Let Gα, α∈I, be nontrivial left (or right) ordered
groups and let ℱ=∏α∈I*Gα be their free product. Since it will be more convenient to work with left order, we assume that Gα, α∈I, are left ordered. It is well known, see [12], and fairly easy to show that there exists a total order ⪯ on ℱ which extends the left orders on groups Gα, α∈I, and which turns ℱ into a left ordered group.
A reduced word W∈ℱ is positive if W≻1. A reduced word W is strongly positive, denoted W≻≻1, if every nonempty suffix of W is positive, i.e., if W≡W1W2 with |W2|>0, then W2≻1. Clearly, a strongly positive word is positive. Note if U,W are strongly positive and UW
is reduced, then UW≻≻1. A word U is (respectively strongly) negative if U-1 is (respectively strongly) positive.
Lemma 4.
Suppose S and T are strongly positive words
and the word S-1T is reduced. Then S-1T is either strongly positive or strongly negative.
Proof.
Let S≡S1S2 and T≡T1T2, where |S2|,|T2|>0. Then S2≻1,
T2≻1 by S,T≻≻1. Since S-1T is reduced, we have S-1T≠1, hence S-1T≻1 or S-1T≺1.
Assume S-1T≻1. Then we have T≻S=S1S2 or S1-1T≻S2≻1, hence, in view of T2≻1, all nonempty suffixes of S-1T are positive. This implies S-1T≻≻1. If S-1T≺1, then, switching S and T, we can show as above
that T-1S≻≻1, hence S-1T is strongly negative.
∎
Lemma 5.
Let W be a reduced word. There exists a factorization W≡U1U2-1 such that
|U1|,|U2|≥0 and each of U1,U2 is either empty or strongly positive.
Proof.
Consider a factorization W≡U1ε1…Ukεk, where, for every j, Uj≻≻1 and εj=±1,
that would be minimal with respect to k. Since for every letter a of W either a≻≻1 or a-1≻≻1, it follows that such a factorization exists and k≤|W|.
Note if εj=εj+1=1, then UjUj+1 is reduced and so
UjUj+1≻≻1. Similarly, if εj=εj+1=-1, then Uj-1Uj+1-1 is reduced and so
Uj+1Uj≻≻1. Hence, it follows from the minimality of k that εj≠εj+1 for all j=1,…,k-1. If now εj=-1 and εj+1=1 for some j, then we can use Lemma 4 and conclude that either Uj-1Uj+1≻≻1 or
Uj+1-1Uj≻≻1, contrary to minimality of k.
Thus, it is proven that either k=1 or k=2 and ε1=1, ε2=-1, as required.
∎
Lemma 6.
Suppose W is a cyclically reduced word. Then there exists a factorization W≡W1W2 such that the cyclic permutation W¯≡W2W1 of W is either strongly positive or strongly negative.
Proof.
By Lemma 5, W≡U1U2-1, where |U1|,|U2|≥0 and Uj≻≻1 if |Uj|>0, j=1,2. Since W2 is reduced, it follows that U2-1U1 is reduced and, by Lemma 4, either U2-1U1≻≻1 or U2-1U1≺≺1. Hence, W¯≡U2-1U1 is a desired cyclic permutation of W.
∎
4 Proving Theorem 1
Let Gα, α∈I, be nontrivial
left (or right) ordered
groups and let ℱ be their free product equipped with a left order ⪯.
Also, fix a total order ≤ on the index set I.
Let Ψ be a finite irreducible 𝒜-labeled graph, where 𝒜=⋃α∈IGα. An edge e∈EΨ is called maximal if there are reduced infinite paths p=p(e)=e1e2…, q=q(e)=f1f2… in Ψ, where ej,fj∈EΨ, such that e=e1, (e1)-=(f1)- is primary, θ(e1)>θ(f1), and, for every index j≥1, both φ(e1…e2j)≺1 and φ(f1…f2j)≺1. Note that the vertices (e1…e2j)+, (f1…f2j)+ are primary and q-1p=…f2-1f1-1e1e2… is a reduced biinfinite path.
Lemma 7.
Suppose Ψ is a finite irreducible A-labeled graph whose Euler characteristic is negative, χ(Ψ)<0. Then
Ψ contains a maximal edge.
Proof.
Since χ(Ψ)<0, Ψ has a connected component Ψ1 with χ(Ψ1)<0. Without loss of generality, we may assume that core(Ψ1)=Ψ1.
It is not difficult to see from χ(Ψ1)<0 and from core(Ψ1)=Ψ1 that, for every pair h,h′∈EΨ1, there is a reduced path p=h…h′ whose first, last edges are h,h′, respectively. Pick a primary vertex o in Ψ1 and two distinct edges t1,u1 with (t1)-=(u1)-=o. Let t,u be some reduced paths such that first edges of t, u are t1,u1, respectively, and t+, u+ have degree >2. Then it follows from the above remark that there are closed paths
r0,s0 starting at t+, u+, respectively, such that the path tr02t-1us02u-1 is reduced.
Since Ψ1 is irreducible and r0,s0 are reduced, it follows φ(r0)≠1, φ(s0)≠1 in ℱ.
Let r,s be some cyclic permutations of the closed paths r0,s0, respectively, that start at some primary vertices and let R=φ(r), S=φ(s) be reduced words. Clearly, R,S are cyclically reduced and |R|=|r|/2>1, |S|=|s|/2>1. By Lemma 6, there are cyclic permutations R¯,S¯ of R,S, respectively, such that R¯εr and S¯εs are strongly positive, where εr,εs∈{±1}. Switching from r0,s0 to r0-1,s0-1, respectively, if necessary, we may assume that εr=εs=-1, i.e., we have that
R¯-1,S¯-1≻≻1.
Let r¯, s¯ denote
cyclic permutations of r,s, respectively, such that φ(r¯)=R¯ and φ(s¯)=S¯. Also, let r¯=r¯1r¯2, s¯=s¯1s¯2 be factorizations of r¯, s¯, respectively, defined by vertices t+, u+, respectively.
Consider two infinite paths starting at o=t-=u- and defined as follows. Let T=tr0+∞ whose prefixes are tr0k, k≥0, and U=us0+∞ whose prefixes are us0ℓ, ℓ≥0. It follows from the definitions that
T starts at t-=o, goes along t to t+ and then cycles
around r0 infinitely many times, in particular, T is reduced. Similarly, U starts at u-=o, goes along u to u+ and then cycles
around s0.
Denote T=t1t2…, where tj∈EΨ1, and U=u1u2…, where uk∈EΨ1. Let T(j1,j2):=tj1…tj2, where j1≤j2, denote the subpath of T that starts at (tj1)- and ends in (tj2)+. It is convenient to set T(j,j-1):={(tj)-} for j≥1.
Similarly, U(j1,j2):=uj1…uj2, where j1≤j2, and U(j,j-1):={(uj)-} if j≥1.
Suppose 2j>|t|+|r¯2|. Then 2j-|t|-|r¯2|>0. Let
m be the remainder of 2j-|t|-|r¯2| when divided by
|r|. Set mr:=m if m>0 and mr:=|r| if m=0.
Note
T(1,2j)=T(1,2j-mr)T(2j-mr+1,2j)
and
φ(T(2j-mr+1,2j))≡R¯3,
where R¯3 is a prefix of
φ(r¯)≡R¯3R¯4 of even length mr>0.
Recall φ(r¯)=R¯ and R¯-1≻≻1,
hence R¯3-1≻1 and R¯3≺1. Note
R¯3=φ(T(2j-mr+1,2j))≺1 implies, by left invariance of the order ⪯, that
(4.1)φ(T(1,2j))≺φ(T(1,2j-mr)).
Now suppose 2j>|u|+|s¯2|. Let
m′ be the remainder of 2j-|t|-|s¯2|>0 when divided by
|s|. Set ms:=m′ if m′>0 and ms:=|s| if m′=0. Then we can derive from S¯-1≻≻1, similar to (4.1), that
(4.2)φ(U(1,2j))≺φ(U(1,2j-ms)).
The comparisons (4.1)–(4.2) prove that a maximal element of the infinite set
(4.3){φ(T(1,2j)),φ(U(1,2k))∣j≥0,k≥1}⊆ℱ
exists and it is the maximal element of the finite set
{φ(T(1,2j)),φ(U(1,2k))∣0≤2j≤|t|+|r¯2|, 0<2k≤|u|+|s¯2|}⊂ℱ.
Let φ(Q(1,2jM)), where jM≥0 and Q∈{T,U}, denote the maximal element of the set (4.3). Note vM=Q(1,2jM)+ is primary. Observe that elements φ(T(1,2j)), φ(U(1,2k)), j≥0, k≥1, in (4.3) are distinct and represent φ-labels of subpaths of the biinfinite path U-1T that connect the primary vertex o=t- to all primary vertices of U-1T along U-1T (or its inverse). If we take another primary vertex v on U-1T and consider the set of labels of subpaths that connect v to primary vertices of U-1T as above, then the resulting set can be obtained from (4.3) by multiplication on the left by φ(h(o,v))-1, where h(o,v)=T(1,2jv) if v=T(1,2jv)+, 2jv≥0, and h(o,v)=U(1,2kv) if v=U(1,2kv)+, 2kv>0. Since left multiplication preserves the order, these remarks imply that the vertex vM=Q(1,2jM)+ defines a factorization of the biinfinite path U-1T=q-1p into infinite paths q, p, where p=e1e2…, q=f1f2…, ej,fj are edges, j≥1, so that, for every j≥1, we have φ(e1e2…e2j)≺1 and φ(f1f2…f2j)≺1. Therefore, if θ(e1)>θ(f1), then e1 is a maximal edge of Ψ and if θ(f1)>θ(e1), then f1 is maximal in Ψ.
∎
Suppose Γ is a finite graph. A set D⊆EΓ of edges is called good (for cutting) if the graph Γ∖(D∪D-1) consists of connected components whose Euler characteristics are 0. Clearly, Γ contains a good edge set if and only if no connected component of Γ is a tree.
Lemma 8.
Suppose Ψ is a finite connected irreducible A-graph with χ(Ψ)<0. Then the set of all maximal edges of Ψ is good (for cutting).
Proof.
Arguing on the contrary, assume that D⊆EΨ is the set of
all maximal edges of Ψ and D is not good. Then the graph Ψ∖(D∪D-1) contains a connected component Ψ1′ with either χ(Ψ1′)<0 or χ(Ψ1′)>0.
If χ(Ψ1′)<0, then the core Ψ1=core(Ψ1′) of Ψ1′ is a finite irreducible 𝒜-graph with χ(Ψ1)<0. By Lemma 7, Ψ1 contains a maximal edge e. However, it follows from the definition that e is also maximal for Ψ, hence, e∈D. This contradiction shows that χ(Ψ1′)>0, hence Ψ1′ is a tree which we denote T.
Let C denote the set that consists of all c∈EΨ such that c+∈VT, c∉ET. It follows
from the definitions that if c∈C, then c∈D or c-1∈D. Since every d∈D
is maximal, there are infinite reduced paths p(d)=e1(d)e2(d)… and q(d)=f1(d)f2(d)… such that
e1(d)-=f1(d)-∈VPΨ,e1(d)=d,θ(e1(d))>θ(f1(d))
and, for every j≥1, φ(e1(d)…e2j(d))≺1 and φ(f1(d)…f2j(d))≺1.
Pick an arbitrary c∈C. Suppose c- is primary. Since d-∈VPΨ if d∈D and c or c-1 is in D, it follows that c∈D. Consider a shortest path of the form h(c):=e1(c)…e2ℓ(c), ℓ≥1, such that either e2ℓ(c)-1∈C or e2ℓ(c)-1∈T and e2ℓ+1(c)-1∈C. Define
σ(c):={e2ℓ(c)-1if e2ℓ(c)-1∈C,e2ℓ+1(c)-1if e2ℓ+1(c)-1∈C.
Since T is a finite tree, such a path h(c) exists and satisfies |h(c)|>0, σ(c)≠c, and φ(h(c))≺1. Note h(c)- and h(c)+ are primary vertices of c, σ(c), respectively, and
h(c)=chT(c)σ(c)-εσ(c), where hT(c) is a subpath of h(c) in T such that
hT(c)-=c+, hT(c)+=σ(c)+, and εσ(c)=1 if σ(c)+ is secondary and εσ(c)=0 if σ(c)+ is primary.
Now assume that c+ is primary. Then we have c-1=dc∈D. Consider a shortest path of the form h(c):=f1(dc)…f2ℓ-2(dc), where ℓ≥1 and if ℓ=1, then h(c):={c+}, such that either f2ℓ-2(dc)-1∈C
or f2ℓ-2(dc)-1∈T (or f2ℓ-2(dc) is undefined if ℓ=1) and f2ℓ-1(dc)-1∈C. Define
σ(c):={f2ℓ-2(dc)-1if f2ℓ-2(dc)-1∈C,f2ℓ-1(dc)-1if f2ℓ-1(dc)-1∈C.
Since T is a finite tree, such a path h(c) exists and satisfies |h(c)|≥0, σ(c)≠c, and φ(h(c))⪯1. In addition, the equality φ(h(c))=1 implies that h(c)={c+},
σ(c)=f1(c)-1 and θ(σ(c))=θ(f1(dc))<θ(e1(dc))=θ(c).
As above, we remark that h(c)- and h(c)+ are primary vertices of c and σ(c), respectively, and
h(c)=hT(c)σ(c)-εσ(c), where hT(c) is in T such that
hT(c)-=c+, hT(c)+=σ(c)+, and εσ(c)=1 if σ(c)+ is secondary
and εσ(c)=0 if σ(c)+ is primary.
Let us summarize. For every c∈C, we have have defined an edge σ(c)∈C, where σ(c)≠c, and hence σ:C→C is a function. Furthermore, there is a reduced path h(c) such that h(c)=cεchT(c)σ(c)-εσ(c), where hT(c) is in T with
hT(c)-=c+, hT(c)+=σ(c)+, εc=1 if c+∈VSΨ and εc=0 if c+∈VPΨ. In addition, we have εσ(c)=1 if σ(c)+∈VSΨ
and εσ(c)=0 if σ(c)+∈VPΨ. Also, φ(h(c))⪯1 and φ(h(c))=1 implies that h(c)={c+}={σ(c)+} and θ(c)>θ(σ(c)). Finally, h(c)-, h(c)+ are primary vertices of c, σ(c), respectively, whence h(c)+=h(σ(c))- for every c∈C.
Since C is finite, there is a cycle c,σ(c),…,σk(c)=c, k≥2, for some c∈C. Consider the closed path
pc=h(c)h(σ(c))…h(σk-1(c)).
Since
φ(h(σj(c)))⪯1 for every j, we obtain that φ(pc)⪯1 and the equality φ(pc)=1 implies φ(h(σj(c)))=1 and h(σj(c))={σj(c)+} for every j.
On the other hand,
pc=cεchT(c)σ(c)-εσ(c)σ(c)εσ(c)hT(σ(c))σ2(c)-εσ2(c)
…σk-1(c)εσk-1(c)hT(σk-1(c))σk(c)-εσk(c)
=cεchT(c)hT(σ(c))…hT(σk-1(c))c-εc
following from σk(c)=c. Since hT(c)hT(σ(c))…hT(σk-1(c)) is a closed path in the tree T, we have φ(pc)=1 in ℱ. Thus,
h(σj(c))={σj(c)+}={σ(c)+} and
θ(σj(c))>θ(σj+1(c)) for every j=0,1,…,k-1, implying
θ(c)>θ(σk(c))=θ(c).
This contradiction completes the proof. ∎
Proof of Theorem 1.
As in Section 2, we consider a finite irreducible 𝒜-graph Ψ(H1,H2) whose connected components correspond to core graphs of subgroups H1∩sH2s-1, s∈S(H1,H2). Without loss of generality,
we may assume that -χ(Ψ(H1,H2))>0.
Let D be the set of all maximal edges in Ψ(H1,H2). It is easy to see from the definitions that if d∈D, then τi(d) is maximal in
Ψoi(Hi), i=1,2. Hence, τi(D)⊆Di, where Di is the set of maximal edges of Ψoi(Hi), i=1,2. By Lemma 8, Di is good for Ψoi(Hi) and it follows from Lemma 3 and the definitions that
r¯(H1,H2)=|D|≤|τ1(D)|⋅|τ2(D)|≤|D1|⋅|D2|=r¯(H1)⋅r¯(H2),
as desired.
∎
