Weighted self-avoiding walks

We study the connective constants of weighted self-avoiding walks (SAWs) on infinite graphs and groups. The main focus is upon weighted SAWs on finitely generated, virtually indicable groups. Such groups possess so-called 'height functions', and this permits the study of SAWs with the special property of being bridges. The group structure is relevant in the interaction between the height function and the weight function. The main difficulties arise when the support of the weight function is unbounded, since the corresponding graph is no longer locally finite. There are two principal results, of which the first is a condition under which the weighted connective constant and the weighted bridge constant are equal. When the weight function has unbounded support, we work with a generalized notion of the 'length' of a walk, which is subject to a certain condition. In the second main result, the above equality is used to prove a continuity theorem for connective constants on the space of weight functions endowed with a suitable distance function.


Introduction
The counting of self-avoiding walks (SAWs) is extended here to the study of weighted SAWs.Each edge is assigned a weight, and the weight of a SAW is defined as the product of the weights of its edges.We study certain properties of the exponential growth rate µ (known as the connective constant) in terms of the weight function φ, including its continuity on the space of weight functions.
The theory largely follows the now established route when the underlying graph G is locally finite.New problems emerge when G is not locally finite, and these are explored in the situation in which G is the complete graph on an infinite group Γ, with weight function φ defined on Γ.
One of the main technical steps in the current work is the proof, subject to certain conditions, of the equality of the weighted connective constant and the weighted bridge constant.This was proved in [8,Thm 4.3] for the unweighted constants on any connected, infinite, quasi-transitive, locally finite, simple graph possessing a unimodular graph height function.This result is extended here to weighted SAWs on finitely generated, virtually indicable groups (see Section 3 and [16,17] for references on virtual indicability).
As explained in [8,11], a key assumption for the definition and study of bridge SAWs is the existence of a so-called 'graph height function' h.Some Cayley graphs (including those of virtually indicable groups) possess such functions, and some do not (see [9]).Our proof of the equality of bridge and connective constants hinges on a combinatorial fact (due to Hardy and Ramanujan, [15], see Remark 3.10) whose application here imposes a condition on the pair (φ, h).This condition generally fails in the non-locally-finite case, but holds if the usual combinatorial definition of the length of a SAW (that is, the number of its edges) is replaced by a generalized length function satisfying a certain property Π( , h), stated in (3.9).In the locally-finite case, the usual graph-distance function invariably satisfies Π( , h).
Certain new problems arise when working with a general length function other than the usual graph-distance, but the reward includes the above desired equality, and also the continuity of the weighted connective constant on the space of weight functions with a suitable distance function.
Here is an informal summary of our principal results.The reader is referred to Sections 2-4 (and in particular, Theorems 3.9 and 4.1) for definitions and formal statements.
Theorem 1.1.Let Γ be an infinite, finitely generated, virtually indicable group, and let φ : Γ → [0, ∞) be a summable, symmetric weight function that spans Γ.If (3.9) holds, then (a) the connective and bridge constants are equal, (b) the connective constant is a continuous function on the space Φ of such weight functions endowed with a suitable distance function.
Similar results are given for locally finite quasi-transitive graphs, see Theorems 2.5 and 2.6.
The principal results of the paper concern weighted SAWs on certain groups including, for example, all finitely generated, elementary amenable groups.The defining property of the groups under study is that they possess so-called 'strong graph height functions' (see [8,10,11] and especially [9]); this is shown in Theorem 3.2 to be essentially equivalent to assuming the groups to be virtually indicable.The algebraic structure of a group Γ plays its role through the interaction between the height function h, the length function , and the weight function φ.Thus the study of weighted SAWs is related to that of long-range models including percolation and the Ising model (see, for example, [1,3] and the references therein).
Weighted SAWs on Z 2 have been considered by Glazman and Manolescu [5,6] in work that extends some results of Duminil-Copin and Smirnov [4] on SAWs on the trangular lattice.Randomly weighted SAWs on grids have been studied by Lacoin [18,19] and Chino and Sakai [2].
Here is a summary of the contents of the article.The asymptotics of weighted self-avoiding walks on quasi-transitive, locally finite graphs are considered in Section 2. The relevance of graph height functions is explained in the context of bridges, and the equality of connective and bridge constants is proved for graphs possessing unimodular graph height functions.In this case, the connective constant is continuous on the space of weight functions with the supremum norm.Proofs are either short or even omitted, since only limited novelty is required beyond [8].
Weighted walks on a class of infinite countable groups are the subject of Sections 3 and 4, namely, on the class of virtually indicable groups.A generalized notion of the length of a SAW is introduced, and a condition is established under which the bridge constant equals the connective constant.The connective constant is shown to be continuous in the weight and length functions.Proofs are largely deferred to Sections 5-7.
We write R for the reals, Z for the integers, and N for the natural numbers.

Weighted walks and bridges on graphs
In this section we consider weighted SAWs on locally finite graphs.The length of a walk is its conventional graph length, that is, the number of its edges.
If φ is H-invariant and H acts quasi-transitively, then A walk on G is called n-step if it traverses exactly n edges (possibly with reversals and repeats).An n-step self-avoiding walk (SAW) on G is an ordered sequence Note that the weight function w = w φ acts symmetrically in that the weight of an edge or SAW is the same irrespective of the direction in which it is traversed.Let Σ n (v) be the set of n-step SAWs starting at v ∈ V , and set Σ n = Σ n (1).For a set Π of SAWs, we write for the total weight of members of Π.
The following is proved as for the unweighted case in [12] (see also [13,19,21]), and the proof is omitted.For H ≤ Aut(G), we let Φ E (H) be the space of functions φ : It was shown in [8,9] (see also the review [11]) how to define 'bridge' SAWs on certain general families of graphs, and how to adapt the proof of Hammersley and Welsh [14] to show equality of the connective constant and the bridge constant.Key to this approach is the following notion of a graph height function.See [9,20] for accounts of unimodularity.

Definition 2.2. A graph height function on
(a) h : V → Z, and h(1) = 0, (b) H ≤ Aut(G) acts quasi-transitively on G, and h is Associated with a graph height function (h, H) are two integers d, r which we define next.Let where u ∼ v means that u and v are neighbours.If H acts transitively, we set r = 0. Assume H does not act transitively, and let r = r(h, H) be the infimum of all r such that the following holds.Let o 1 , o 2 , . . ., o M be representatives of the orbits of H.For i = j, there exists v j ∈ Ho j such that h(o i ) < h(v j ), and a SAW ν i,j := ν(o i , v j ) from o i to v j , with length r or less, all of whose vertices x, other than its endvertices, satisfy h(o i ) < h(x) < h(v j ).We fix such SAWs ν i,j , and we set ν i,i = {o i }.These SAWs will be used in Sections 5 and 6.Meanwhile, set (2.4) φ ν = min φ(e) : e ∈ ν i,j for some i, j , and (2.5) θ i,j = w(ν i,j ), where θ i,i := 1.If H acts transitively, we set φ ν = θ min = θ max = 1.Some properties of r and d have been established in [8,Prop. 3.2].
We turn to so-called half-space walks and bridges.Let G = (V, E) ∈ G have a graph height function (h, H), and let and we write H n (v) for the set of half-space walks π with initial vertex v.We call π a bridge if and a reversed bridge if (2.6) is replaced by The span of a SAW π is defined as Let β n (v) be the set of n-step bridges π from v, and let It is easily seen (as in [14]) that from which we deduce the existence of the bridge constant and furthermore where r = r(h, H) and φ ν are given after (2.3). Theorem Proof of Proposition 2.4.Assume G has graph height function (h, H).If G is transitive, the claim is trivial by (2.9) and (2.10), so we assume G is quasi-transitive but not transitive.Choose x ∈ V such that w(β n+r (x)) = w(β n+r ).Let v ∈ V , and let v have type o j and x type o i .Let ν(o i , v j ) be given as above (2.4).The length l(o i , v j ) of ν(o i , v j ) satisfies l(o i , v j ) ≤ r.Find η ∈ H such that η(v j ) = v, and let ν(x, v) = η(ν(o i , v j )).Let l = l(o i , v j ) if i = j, and l = 0 otherwise.Then, φ r ν w(β n (v)) ≤ φ r−l ν w(β n+l (x)) ≤ w(β n+r (x)) = w(β n+r ), and (2.12) follows by (2.10).The limit (2.11) follows by (2.7) and (2.9).
Proof of Theorem 2.6.We shall show continuity at the point from which (2.14) follows.

Weighted walks on infinite groups
We consider weighted SAWs on groups in this section.If the weight function has bounded support, the relevant graph is locally finite, and the methods and conclusions of the last section apply.New methods are needed if the support is unbounded, and it turns out to be useful to consider a different measure of the length of a walk.We shall consider groups that support graph height functions.
A group Γ is called indicable if there exists a surjective homomorphism If this holds, we call Γ virtually H-indicable (or, sometimes, virtually (H, F )-indicable).(See, for example, [16,17] for information on virtual indicability.) We remind the reader of the definition of a strong graph height function, as derived from [10,Defn 3.4].Definition 3.1.A (not necessarily locally finite) Cayley graph G = (Γ, F ) of a finitely generated group Γ is said to have a strong graph height function (h, H) if the following two conditions hold.
(a) H Γ acts on Γ by left-multiplication, and The group properties of: (i) virtual indicability, and (ii) the possession of a strong graph height function, are equivalent in the sense of the next theorem.In the context of this theorem, H acts freely on the Cayley graph G, and is therefore unimodular.Hence, the functions (h, H) are unimodular graph height functions on G.The proofs of this and other results in this section may be found in Section 5.
Here is some notation.Let Γ be finitely generated and virtually (H, F )-indicable.It is shown in [9,Sect. 7] that, for a given locally finite Cayley graph G, there exists a unique harmonic extension ψ of F , and that ψ takes rational values.It is then shown how to construct a harmonic, strong graph height function for G of the form (h, H).The latter construction is not unique.For given (Γ, H, F, G), we write h = h F for a given such function.
By Theorems 2.5 and 3.2, the weighted bridge constant equals the weighted connective constant for any locally finite, weighted Cayley graph of a finitely generated, virtually indicable group.Here is an example of a class of such groups.We turn to weight functions and SAWs.Let Γ be virtually H-indicable, and let φ : Γ → [0, ∞) satisfy φ(1) = 0 and be symmetric in that The support of φ is the set supp(φ) := {γ : φ(γ) > 0}.We call φ summable if satisfies w(Γ) < ∞.We write When φ is summable, we have that We misuse notation by setting φ(α, γ) = φ(α −1 γ) for α, γ ∈ Γ.We consider SAWs on the complete graph K = (Γ, E) with vertex-set Γ and weights w = w φ as in (2.2).We say that φ spans Γ if, for η, γ ∈ Γ, there exists a SAW on K from η to γ with strictly positive weight.This is equivalent to requiring that supp(φ) generates Γ.Let Φ be the set of functions φ : Γ → [0, ∞) that are symmetric, summable, and which span Γ, and let Φ bnd be the subset of Φ containing functions φ with bounded support.Functions in Φ are called weight functions.
Let φ ∈ Φ.Since we shall be interested only in SAWs π with strictly positive weights w(π), we shall have use for the subgraph Proof.This is a corollary of Theorem 3.2(a), since supp(φ) is a generator-set of Γ.
Let Γ be virtually (H, F )-indicable, and let (h F , H) be the strong graph height function constructed after Theorem 3.2.A SAW π on K is a bridge (with respect to h F ) if (2.6) holds.Let φ ∈ Φ and ∈ Λ.For v ∈ V , let Σ(v) (respectively, B(v)) be the set of SAWs (respectively, bridges) on G starting at v, and abbreviate Σ(1) = Σ.For c > 0, let We shall abbreviate σ m,c (v) (respectively, β m,c (v)) to σ m,c (v) (respectively, β m,c (v)) when confusion is unlikely to arise.Since K is transitive and is Γ-invariant, σ m,c (v) does not depend on the choice of v, and we write σ m,c for its common value.In contrast, β m,c (v) may depend on v since the height function h is generally only Hdifference-invariant.Although we shall take limits as m → ∞ through the integers, it will be useful sometimes to allow m to be non-integral in (3.6).
(a) For all sufficiently large c, the limits exist and are independent of the choice of c (and of the choice of v, in the latter case).They are called the (weighted) connective constant and (weighted) bridge constant, respectively.
We have no proof in general of the full convergence w φ (σ m,c ) 1/m → µ φ, , but we shall see in Theorem 3.9 that this holds subject to an additional condition.
Theorem 3.7(a) assumes a lower bound on the length c of the intervals in (3.6).That some lower bound is necessary is seen by considering Example 3.6 with a weight function φ every non-zero value of which has the form 2 −s for some integer s ≥ 1.When this holds, every -length is a multiple of 2, implying that σ m,c , β m,c = ∅ if m is odd and c < 1.The required lower bounds on c are discussed after the statement of Proposition 5.4; see (5.3).
We introduce next a condition on the triple (φ, , h F ). Let Γ be virtually (H, F )indicable with strong graph height function (h F , H), and let φ ∈ Φ.For ∈ [1, 2) and C > 0, let Λ = Λ (C, φ) be the subset of Λ containing functions satisfying the following Hölder condition for the height function: Here is our first main theorem, proved in Section 6.Once again, we require c ≥ A where A will be given in (5.3).A sufficient condition for (3.9) is presented in Lemma 5.2.
Remark 3.10 (Condition (3.9)).Some condition of type (3.9) is necessary for the proof that µ φ, = β φ, (presented in Section 6) for the following reason.A key estimate in the proof of [14] concerning the bridge constant is the classical Hardy-Ramanujan [15] estimate of exp π n/3 for the number of ordered partitions of an integer n.It is important for the proof that this number is e o(n) .Inequality (3.9) implies that the aggregate height difference along a SAW π has order no greater than n := ( (π)) with < 2, which has exp o( (π)) ordered partitions.(See Proposition 5.6.) Remark 3.11 (Working with graph-distance).Let us work with the graph-distance of Example 3.5, writing Σ n and β n as usual.Then w(Σ n ) is sub-multiplicative, whence the limit µ = lim n→∞ w(Σ n ) 1/n exists.By Theorem 3.
It is not hard to see the converse, as follows.Let π = (π 0 = 1, π 1 , . . ., π n ) be a SAW with w(π) > 0, and write We have no general proof of the equality of µ and the bridge limit β when supp(φ) is unbounded, although this holds subject to (3.9) with taken as the graph-distance of Example 3.5.
Example 3.12 (Still working with graph-distance).Let φ ∈ Φ and take to be graph-distance on K φ , as in Example 3.5.By Theorem 3.7(b), the limit µ φ, = lim m→∞ w(σ m,c ) 1/m exists, and it is easy to see that one may take any c > 0.Here is an example in which K φ is not locally finite.Let Γ = Z 2 , and let Since µ φ, is independent of the choice of triple (h F , H, F ), so is β φ, .

Continuity of the connective constant
A continuity theorem for connective constants is proved in this section.We work on the space Φ × Λ with distance function where the suprema are over the set supp(φ) ∪ supp(ψ).This may be compared with (2.13).
The following continuity theorem is our second main theorem, and it is proved in Section 7.For C, W > 0 and ∈ [1, 2), we write Φ • Λ (C, W ) for the space of all pairs (φ, ) satisfying φ ∈ Φ, ∈ Λ (C, φ), and µ φ, ≤ W . Section 7 contains also a proposition concerning the effect on a connective constant µ φ, of truncating the weight function φ; see Proposition 7.4.

Proofs of Theorems 3.2 and 3.7
We prove next the aforesaid theorems together with some results in preparation for the proof in Section 6 of Theorem 3.9.We suppress explicit reference to φ and except where necessary to avoid ambiguity.Lemma 5.1.Let Γ be finitely generated and let G be a (not necessarily locally finite) Cayley graph of Γ with generator-set S = (s i : i ∈ I).There exists a locally finite Cayley graph G of Γ with (finite) generator-set S ⊆ S.
Proof.Let S 0 be a finite generator-set of Γ.Each s ∈ S 0 can be expressed as a finite product of the form s = {ψ : ψ ∈ T s } with T s ⊆ S. We set S = s∈S 0 T s .
Proof of Theorem 3.2.(a) Let G be a Cayley graph of Γ with generator-set S. By Lemma 5.1, we can construct a locally finite Cayley graph G of Γ with respect to a finite set S ⊆ S of generators.Since H acts freely and is therefore unimodular, we may apply [9,Thm 3.4] to deduce that G has a strong graph height function of the form (h, H).Since G is a subgraph of G with the same vertex set, and h is H-difference-invariant on G , it is H-difference-invariant on G also.
It remains to check that each vertex v ∈ Γ has neighbours u, w in G with h(u) < h(v) < h(w).This holds since there exist s 1 , s 2 ∈ S ⊆ S such that h(vs 1 ) < h(v) < h(vs 2 ).

(b) Let G be a Cayley graph of Γ with a strong graph height function (h, H). Then H Γ and [Γ
There follows a sufficient condition for the condition (3.9) of the principal Theorem 3.9.
Lemma 5.2.Let Γ be finitely generated and φ ∈ Φ.Let G be the Cayley graph with generator-set supp(φ), and let G be as in Lemma 5.1.Write δ for graph-distance on G .Let (h F , H) be the strong graph height function described afterTheorem 3.2.If there exists C 1 > 0 and ∈ [1, 2) such that then there exists C > 0 such that (3.9) holds.
Proof.Since (h F , H) is a graph height function on the locally finite graph G , by (2.3) there exists d ∈ (0, ∞) such that Inequality (3.9) follows by (5.1) with C = dC − 1 .Lemma 5.3.Let Γ be finitely generated and virtually H-indicable, where H = Γ, and let φ ∈ Φ.Let K φ = (Γ, E φ ) be as in (3.3), and let (h, H) be a strong graph height function on K φ .Let o i be a representative of the ith orbit of H.There exists an integer s = s(h, H, φ) ∈ N such that the following holds.For i = j, there exists v j ∈ Ho j such that h(o i ) < h(v j ), and a SAW ν i,j = ν(o i , v j ) of G from o i to v j with -length s or less, each of whose vertices x other than its endvertices satisfy h(o i ) < h(x) < h(v j ).
Proof.This extends to K φ the discussion above (2.4).Let G be a locally finite Cayley graph of Γ with respect to a finite generator-set S ⊆ supp(φ), as in Lemma 5.1.By [8, Prop.3.2], there exists 0 < r < ∞ such that, for i = j, there exists v j ∈ Ho j such that h(o i ) < h(v j ), and a SAW ν i,j = ν(o i , v j ) on G from o i to v j with graph-length r or less.Furthermore, each of the vertices x of ν(o i , v j ) other than its endvertices satisfy h(o i ) < h(x) < h(v j ).
Since G is a subgraph of G, a SAW on G is also a SAW on G.Moreover, since H acts quasi-transitively, and every edge of G has finite -length, we can find s ∈ (0, ∞) such that the claim holds.
In advance of the proof, we introduce some notation that will be useful later in this work.Let G = (Γ, E ) be the Cayley graph generated by the finite S ⊆ supp(φ) of Lemma 5.1 applied to G := K φ .By Theorem 3.2(a) and its proof, there exists (h F , H) which is a strong graph height function of both K φ and G .
Since h F is a graph height function on G , for v ∈ Γ, there exists an edge e v = v, vα v ∈ E such that h F (vα v ) > h F (v) (and also φ(α v ) > 0 by definition of G ).We call the edge e v the extension at v. Write noting that ψ, a > 0 and A < ∞ since h F is H-difference-invariant and H acts quasi-transitively.Let (5.4) Lemma 5.5.For q ≥ c − A ≥ 0, there exists Proof.Let π ∈ β p,c (v).We may extend π by adding progressive extensions at the final endpoint.After some number r of such extensions, we achieve a bridge π contributing to β p+q,A .Note that w(π ) ≥ ψ r w(π), and r ≤ q/a .Each such π occurs no more than c/a times in this construction.The claimed inequality holds with C 1 = c/a ψ − q/a .
Proof of Lemma 5.4.By concatenation as usual, By Lemma 5.5, there exists C = C(ψ, A, a) such that This holds for all v, whence (5.5) We deduce (by subadditivity, as in [7, eqn (8.38)], for example) the existence of the limit In particular, by (5.4), By (5.5), (5.7) We show next that (5.2) (with c = A) is valid for all v ∈ Γ.Let s be as in Lemma 5.3, and let x, v ∈ Γ have types o i , o j , respectively, where i = j.Any b v ∈ β m,A (v) may be prolonged 'backwards' by a translate αν(o i , v j ), for suitable α ∈ H, to obtain a bridge b x from x with weight w(b x ) = θ i,j w(b v ), where θ i,j = w(ν(o i , v j )) as in (2.5).By Lemma 5.5, there exists If i = j, this holds with θ i,j = 1.
Proof of Theorem 3. (c) It is trivial that β φ, ≤ µ φ, , and it was noted at the end of the proof of part (a) that β φ, > 0. Let Σ n be the set of n-step SAWs from 1.By (3.5), as required.
We shall introduce several constants, denoted D, which depend on certain parameters as specified.The single character D is used repeatedly for economy of notation; the value of such D can vary between appearances.We recall the constants θ min , θ max of (2.5), r as after (2.3), s of Lemma 5.3, and ψ, A, a of (5.3).Proposition 6.1.There exist constants D = D(ψ, A, a, θ min , θ max , r, s), which are continuous functions of ψ, A, a, θ min , θ max , such that Proposition 6.2.There exist constants D = D(ψ, A, a, θ min , θ max , r, s), which are continuous functions of ψ, A, a, θ min , θ max , such that where N is given in (6.1).
Proof of Proposition 6.1.
have graph-length n and -length (π).Let n 0 = 0, and for j ≥ 1, define S j = S j (π) and n j = n j (π) recursively as follows: and n j is the largest value of t at which the maximum is attained.The recursion is stopped at the smallest integer k = k(π) such that n k = n, so that S k+1 and n k+1 are undefined.Note that S 1 is the span of π and, more generally, S j+1 is the span of the SAW π j+1 := (π n j , π n j +1 , . . ., π n j+1 ).Moreover, each of the subwalks π j+1 is either a bridge or a reversed bridge.We observe that (a 1 , a 2 , . . ., a k ) be the set of half-space walks π from v ∈ Γ with -length satisfying (π) ∈ [m, m+c), and such that k(π) = k, S 1 (π) = a 1 , . . ., S k (π) = a k , and n k (π) = n (and hence S k+1 is undefined).We abbreviate B v m,A = B v m .In particular, B v m (p) is the set of bridges π from v with span p and -length (π) ∈ [m, m + A).
Recall the weight θ i,j of the SAW ν i,j = ν(o i , v j ) as after (2.3).Let s i,j be the -length of ν i,j , and (6.2) as in Lemma 5.3.We shall perform surgery on π to obtain a SAW π satisfying for some i, j depending on π.Note that the subscripts in (6.3) may be non-integral.
The new SAW π is constructed in the following way.Suppose first that k ≥ 3.In the following, we use the fact that H acts on Γ by left-multiplication.
σ ρ σ ρ Figure 6.1.Two images of ν i,j are introduced in order to make the required connections.These extra connections are drawn as black straight-line segments, and each has weight θ i,j , -length s i,j , and span δ i,j .
We consider next the multiplicities associated with the map π → π .The argument in the proof of [8,Sect. 7] may not be used directly since the graph K φ is not assumed locally finite.The argument required here is however considerably simpler since we are working with Cayley graphs.
Let (π(1), σ , ρ ) be such an admissible representation of π .Since H acts by leftmultiplication, there exists a unique α 2 ∈ H such that α 2 π n equals the final endpoint of ρ .It follows that ρ = α −1 2 ρ .By a similar argument, there exists a unique choice for σ that corresponds to the given representation.
Proof of Theorem 3.9.That µ φ, = β φ, follows immediately from Proposition 6.We begin with a proposition, and then prove Theorem 4.1.The section ends with a proof that, as the truncation of a weight function is progressively removed, the connective constant converges to its original value (see (7.13) and Proposition 7.4).
The set of such pairs (φ, ) is denoted C. Proof.Since ∈ Λ (C, φ) and φ η ≤ φ, we have by (3.9) that ∈ Λ (C, φ η ) for η < M .Therefore, Theorem 3.9 may be applied to the pairs (φ η , ).Let ζ ∈ (0, M ), so that φ ζ spans Γ.Let K φ ζ be the locally finite Cayley graph of Γ on the edges e for which φ ζ (e) > 0 (see Lemma 5.1), with corresponding strong graph height function (h, H), and let ψ, A, a be as in (5.3) with φ replaced by φ ζ .Working on the graph K φ ζ with the weight function φ ζ , we construct the paths ν i,j as before (2.4), and we shall stay with these particular paths in the rest of the proof.Since φ η is constant on these paths for η ≤ ζ, the values of ψ, A, a, θ min , θ max , r, s are unchanged for η ≤ ζ.
We shall write w φ since we shall work with more than one weight function.By (7.13)-(7.15),w φ (σ m,A ) = w φ ρ (σ m,A ).(7.16) Since φ ρ ≤ φ, we have by (3.9) that ∈ Λ (C, φ ρ ).By Proposition 6.2 applied to φ ρ , (7.17We omit this proof since it lies close to those of Theorem 3.9 and [8, Thm 4.3].Here is a summary of the differences.Theorem 2.5 differs from [8,Thm 4.3] in that edges are weighted, and this difference is handled in very much the same manner as in the proof of Theorem 3.9.On the other hand, Theorem 2.5 differs from Theorem 3.9 in that the underlying graph need not be a Cayley graph.This is handled as in the proof of [8,Thm 4.3], where the unimodularity of the automorphism groups in question is utilized.

Theorem 3 . 2 .
Let Γ be a finitely generated group.(a) If Γ is virtually (H, F )-indicable, any (not necessarily locally finite) Cayley graph G = (Γ, F ) of Γ possesses a strong graph height function of the form (h, H).(b) If some Cayley graph G of Γ possesses a strong graph height function, denoted (h, H), then Γ is virtually H-indicable.

Proposition 3 . 4 .
Let Γ be finitely generated and virtually H-indicable, and let φ ∈ Φ.The graph K φ of (3.3) possesses a strong graph height function of the form (h, H).
the Cayley graph of Γ in which (m, n) and (m , n ) are joined by an edge if and only if either m = m or (m , n ) = (m ± 1, n).Define the height function by h(m, n) = m, and note that (3.9) holds for suitable C. Remark 3.13.Let Γ be finitely generated and virtually (H, F )-indicable, and let φ ∈ Φ and ∈ Λ (C, φ) for some ∈ [1, 2) and C > 0.