Self-avoiding walks and amenability
The connective constant of an infinite transitive graph is the exponential growth rate of the number of self-avoiding walks from a given origin. The relationship between connective constants and amenability is explored in the current work.
Various properties of connective constants depend on the existence of so-called ‘graph height functions’, namely: (i) whether is a local function on certain graphs derived from , (ii) the equality of and the asymptotic growth rate of bridges, and (iii) whether there exists a terminating algorithm for approximating to a given degree of accuracy.
In the context of amenable groups, it is proved that the Cayley graphs of infinite, finitely generated, elementary amenable groups support graph height functions, which are in addition harmonic. In contrast, the Cayley graph of the Grigorchuk group, which is amenable but not elementary amenable, does not have a graph height function.
In the context of non-amenable, transitive graphs, a lower bound is presented for the connective constant in terms of the spectral bottom of the graph. This is a strengthening of an earlier result of the same authors. Secondly, using a percolation inequality of Benjamini, Nachmias, and Peres, it is explained that the connective constant of a non-amenable, transitive graph with large girth is close to that of a regular tree. Examples are given of non-amenable groups without graph height functions, of which one is the Higman group.
Key words and phrases:Self-avoiding walk, connective constant, Cayley graph, amenable group, elementary amenable group, Grigorchuk group, Higman group, Baumslag–Solitar group, graph height function, group height function, harmonic function, spectral radius, spectral bottom
2010 Mathematics Subject Classification:05C30, 20F65, 60K35, 82B20
- A Background, and summary of results
- B Results for amenable groups
- C Results for non-amenable graphs
- D Remaining proofs
Part A Background, and summary of results
A self-avoiding walk on a graph is a path that visits no vertex more than once. The study of the number of self-avoiding walks of length from a given initial vertex was initiated by Flory  in his work on polymerization, and this topic has acquired an iconic status in the mathematics and physics associated with lattice-graphs. Hammersley and Morton  proved in 1954 that, if is vertex-transitive, there exists a constant , called the connective constant of , such that as . This result is important not only for its intrinsic value, but also because its proof contained the introduction of subadditivity to the theory of interacting systems. Subsequent work has concentrated on understanding polynomial corrections in the above asymptotic for (see, for example, [3, 36]), and on finding exact values and inequalities for connective constants (for example, [10, 18]).
There are several natural questions about connective constants whose answers depend on whether or not the underlying graph admits a so-called graph height function. The first of these is whether is a continuous function of the graph (see [4, 19]). This so-called locality question has received serious attention also in the context of percolation and other disordered systems (see [5, 37, 39]), and has been studied in recent work of the current authors on general transitive graphs, , and also on Cayley graphs of finitely generated groups, . Secondly, when has a graph height function, one may define bridge self-avoiding walks on , and show that their numbers grow asymptotically in the same manner as (see ). The third such question is whether there exists a terminating algorithm to approximate within any given (non-zero) margin of accuracy (see [19, 20]).
Roughly speaking, a graph height function on is a non-constant function whose increments are invariant under the action of a finite-index subgroup of automorphisms (a formal definition may be found at Definition 3.1). It is, therefore, useful to know which transitive graphs support graph height functions.
A method for constructing graph height functions on a certain class of transitive graphs is described in , and the question is posed there of deciding whether all transitive graphs support graph height functions. A rich source of interesting examples of transitive graphs is provided by Cayley graphs of finitely generated groups, as studied in . It is proved there that the Cayley graphs of finitely generated, virtually solvable groups support graph height functions, which are in addition harmonic. The question is posed of determining whether or not the Cayley graph of the Grigorchuk group possesses a graph height function.
We are concerned here with the relationship between connective constants and amenability, and we present results for both amenable and for non-amenable graphs. Since these results are fairly distinct, we summarize them here under the two headings of amenable groups and non-amenable graphs.
1.2. Amenable groups
This part of the current work has two principal results, one positive and the other negative.
(Theorem 5.1) It is proved that the Cayley graph of the Grigorchuk group does not support a graph height function. This answers in the negative the above question of  (see also [22, Sect. 5]). Since the Grigorchuk group is amenable (but not elementary amenable), possession of a graph height function is not a characteristic of amenable groups. This is in contrast with work of Lee and Peres, , who have studied the existence of non-constant, Hilbert space valued, equivariant harmonic maps on amenable graphs.
1.3. Non-amenable graphs
In earlier work , it was shown that the connective constant of a transitive, simple graph with degree satisfies
and it was asked whether or not the lower bound is sharp. In the first of the following three results, this is answered in the negative for non-amenable graphs.
Relevant notation for graphs, groups, and self-avoiding walks is summarized in Section 2, and three different types of height functions are explained in Section 3. The class of elementary amenable groups is described in Sections 4 and 9. The Grigorchuk group is defined in Section 5 and the non-existence of graph height functions thereon is given in Theorem 5.1. The improved lower bound for for non-amenable is presented at Theorem 6.2, and the remark about non-amenable graphs with large girth at Theorem 7.1. The Higman group is discussed in Section 8. Proofs of theorems appear either immediately after their statements, or are deferred to self-contained sections at the end of the article.
2. Graphs, groups, and self-avoiding walks
The graphs in this paper are simple, in that they have neither loops nor multiple edges. The degree of vertex is the number of edges incident to . We write for neighbours and , for the neighbour set of , and (respectively, ) for the set of edges incident to (respectively, between and ). The graph is locally finite if for . An edge from to is denoted when undirected, and when directed from to . The girth of is the infimum of the lengths of its circuits. The infinite -regular tree crops up periodically in this paper.
The automorphism group of is denoted . The subgroup is said to act transitively on if, for , there exists with . It acts quasi-transitively if there exists a finite subset such that, for , there exists and such that . The graph is said to be (vertex-)transitive if acts transitively on .
Let be the set of infinite, locally finite, connected, transitive, simple graphs, and let . The edge-isoperimetric constant is defined here as
We call amenable if and non-amenable otherwise. See [34, Sect. 6] for an account of graph amenability.
2.2. Self-avoiding walks
Let . We choose a vertex of and call it the origin, denoted . An -step self-avoiding walk (SAW) on is a walk containing edges no vertex of which appears more than once. Let be the set of -step SAWs starting at , with cardinality . We have in the usual way (see [25, 36]) that
whence the connective constant
A SAW is called extendable if it is the initial portion of an infinite SAW on . (An extendable SAW is called ‘forward extendable’ in .)
Let be a group with generator set satisfying and , where is the identity element. We shall assume that , while noting that this was not assumed in . We write with a set of relators (or relations, when convenient). Such a group is called finitely generated, and is called finitely presented if, in addition, .
The Cayley graph of the presentation is the simple graph with vertex-set , and an (undirected) edge if and only if for some . Thus, our Cayley graphs are simple graphs. See [2, 34] for accounts of Cayley graphs, and  of geometric group theory.
3. Height functions
It was shown in  that graphs supporting so-called ‘graph height functions’ have (at least) three properties:
one may define the concept of a ‘bridge’ SAW on , as in ,
the exponential growth rate for counts of bridges equals the connective constant ,
there exists a terminating algorithm for determining to within any prescribed (strictly positive) degree of accuracy.
Several natural sub-classes of contain only graphs that support graph height functions, and it was asked in  whether or not every supports a graph height function. This question will be answered in the negative at Theorems 5.1 and 8.1, where it is proved that neither the Grigorchuk nor Higman graphs possess a graph height function. Arguments for proving the non-existence of graph height functions may be found in Section 10.
We review the definitions of the two types of height functions, and introduce a third type. Let , and let . A function is said to be -difference-invariant if
Definition 3.1 ().
A graph height function on is a pair , where acts quasi-transitively on and , such that:
for , there exist such that .
We turn to Cayley graphs of finitely generated groups. Let be a finitely generated group with presentation . As in Section 2, we assume and .
A group height function on (or on a Cayley graph of ) is a function such that:
, and is not identically zero,
if with , then ,
the values are such that, if is a representation of the identity with , then .
A necessary and sufficient condition for the existence of a group height function is given in [22, Thm 4.1]. In the language of group theory, this condition amounts to requiring that the first Betti number is strictly positive. It was recalled in [22, Remark 4.2] that (when the non-zero , , are coprime) a group height function is simply a surjective homomorphism from to .
We introduce a third type of height function, which may be viewed as an intermediary between a graph height function and group height function.
For a Cayley graph of a finitely generated group , we say that the pair is a strong graph height function of the pair if
acts on by left multiplication, and , and
is a graph height function.
It is evident that a group height function (of ) is a strong graph height function of the form , and a strong graph height function is a graph height function. The assumption in (a) above of the normality of is benign, as in Remark 3.2.
We recall the definition of a harmonic function. A function is called harmonic on the graph if
It is an exercise to show that any group height function is harmonic.
Part B Results for amenable groups
4. Elementary amenable groups
The class of elementary amenable groups was introduced by Day in 1957, , as the smallest class of groups that contains the set of all finite and abelian groups, and is closed under the operations of taking subgroups, and of forming quotients, extensions, and directed unions. Day noted that every group in is amenable (see also von Neumann ). An important example of an amenable but not elementary amenable group was described by Grigorchuk in 1984, . Grigorchuk’s group is important in the study of height functions, and we return to this in Section 5.
Let be the set of infinite, finitely generated members of .
Let . Any locally finite Cayley graph of admits a harmonic, strong graph height function.
We prove a slightly stronger version of this at Theorem 9.1, using transfinite induction. The class includes all virtually solvable groups, and thus Theorem 4.1 extends [22, Thm 5.1]. Since any finitely generated group with polynomial growth is virtually nilpotent, , and hence lies in , its locally finite Cayley graphs admit harmonic graph height functions.
5. The Grigorchuk group
The (first) Grigorchuk group is an infinite, finitely generated, amenable group that is not elementary amenable. We show in Theorem 5.1 that there exists a locally finite Cayley graph of the Grigorchuk group with no graph height function. This answers in the negative Question 3.3 of  (see also [22, Sect. 3]).
Here is the definition of the group in question (see [12, 13, 15]). Let be the rooted binary tree with root vertex . The vertex-set of can be identified with the set of finite strings having entries , , where the empty string corresponds to the root . Let be the subtree of all vertices with root labelled .
Let be the automorphism group of , and let be the automorphism that, for each string , interchanges the two vertices and .
Any may be applied in a natural way to either subtree , . Given two elements , we define to be the automorphism on obtained by applying to and to . Define automorphisms , , of recursively as follows:
where is the identity automorphism. The Grigorchuk group is defined as the subgroup of generated by the set .
The Cayley graph of the Grigorchuk group with generator set satisfies:
admits no graph height function,
for with finite index, any -difference-invariant function on is constant on each orbit of .
The proof of Theorem 5.1 is given in Section 11. In the preceding Section 10, two approaches are developed for showing the absence of a graph height function within particular classes of Cayley graph. In the case of the Grigorchuk group, two reasons combine to forbid graph height functions, namely, its Cayley group has no automorphisms beyond the action of the group itself, and the group is a torsion group in that every element has finite order.
Since the Grigorchuk group is amenable, Theorems 4.1 and 5.1 yield that: within the class of infinite, finitely generated groups, every elementary amenable group has a graph height function, but there exists an amenable group without a graph height function. The Grigorchuk group is finitely generated but not finitely presented, [13, Thm 6.2].
We ask if there exists an infinite, finitely presented, amenable group with a Cayley graph having no graph height function. A natural candidate might be the group of [14, Thm 1], with
This finitely presented, amenable HNN-extension of the Grigorchuk group is not elementary amenable. However, since it contains the free group generated by the stable letter , it possesses a group height function. More precisely, the function
defines a group height function.
Part C Results for non-amenable graphs
6. Connective constants of non-amenable graphs
Let have degree . It was proved in [21, Thm 4.1] that
The upper bound is achieved by the -regular tree . It is unknown if the lower bound is sharp for simple graphs. This lower bound may however be improved for non-amenable graphs, as follows.
Let be the transition matrix of simple random walk (SRW) on , and let be the identity matrix. The spectral bottom of is defined to be the largest with the property that, for all ,
Let have degree . Let be the transition matrix of SRW on , and the spectral bottom of . The connective constant satisfies
The improvement in the lower bound for is strict if and only if , which is to say that is non-amenable. It is standard (see [34, Thm 6.7]) that
where is the girth of , is the -regular tree, and
The spectral bottom (and therefore the spectral radius, also) is not a continuous function of in the usual graph metric (see [19, Sect. 5]). This follows from [40, Thm 2.4], where it is proved that, for all pairs with and , there exists a group with polynomial growth whose Cayley graph is -regular with girth exceeding . Since is amenable, we have , whereas is given by (6.5).
Proof of Theorem 6.2.
This is achieved by a refinement of the argument used to prove [21, Thm 4.1], and we shall make use of the notation introduced in that proof.
Let , and let be an extendable -step SAW of . For convenience, we augment with a mid-edge incident to and not lying on the edge . Let be the set of oriented edges such that: (i) , , and (ii) the (non-oriented) edge does not lie in . Note that
Each (oriented) edge in is coloured either red or blue according to the following rule. For , let be the sub-path of joining to . The edge is coloured red if is not an extendable SAW, and is coloured blue otherwise. By (6.6), the number (respectively, ) of blue edges (respectively, red edges) satisfies
We shall make use of the following lemma.
The number satisfies
Proof of Lemma 6.4.
An edge is said to be finite if lies in a finite component of , and infinite otherwise. If is red, then is necessarily finite. Blue edges, on the other hand, may be either finite or infinite.
It was explained in the proof of [21, Thm 4.1] that there exists an injection from the set of red edges to the set of blue edges with the property that, if is red, and , then and lie in the same component of . Since is finite, so is . It follows that
where is the number of red edges, and is the number of infinite blue edges.
Let be a SRW on , and let denote the law of started at . For , let
By [5, Lemma 2.1] with ,
If is finite, then . By (6.10),
The last probability depends on the graph , and it is a maximum when is the -regular tree (since is the universal cover of ). Therefore, it is no greater than multiplied by the probability that a random walk on , which moves rightwards with probability and leftwards with probability , never visits having started at . By, for example, [23, Example 12.59],
7. Graphs with large girth
Benjamini, Nachmias, and Peres showed in [5, Thm 1.1] that the critical probability of bond percolation on a -regular, non-amenable graph with large girth is close to that of the critical probability of the -regular tree . Their main result implies the following.
We recall from Remark 6.1 that if and only if is non-amenable. Theorem 7.1 does not, of itself, imply that is continuous at , since is not continuous at (in the case when is even, see Remark 6.3). For continuity at , it would suffice that is bounded away from on a neighbourhood of . By (6.3), this is valid within any class of graphs whose edge-isoperimetric constants (2.1) are bounded uniformly from . See also [19, Thm 5.1].
8. The Higman group
The Higman group of  is the infinite, finitely presented group with presentation where
This group is interesting since it has no proper normal subgroup with finite index, and the quotient of by its maximal proper normal subgroup is an infinite, finitely generated, simple group. By [22, Thm 4.1(b)], has no group height function. The above two reasons conspire to forbid graph height functions.
The Cayley graph of the Higman group has no graph height function.
A further group of Higman type is given as follows. Let be as above, and let be the finitely presented group with
Note that is infinite and non-amenable, since the subgroup generated by the set is a free group (as in the corresponding step for the Higman group at [29, pp. 62–63]).
The Cayley graph of the above group has no graph height function.
Part D Remaining proofs
9. Proof of Theorem 4.1
We shall prove the following stronger form of Theorem 4.1.
Let . There exists a normal subgroup with such that any locally finite Cayley graph of possesses a harmonic, strong graph height function of the form .
Whereas every member of has a proper, normal subgroup with finite index, it is proved in  that there exist amenable simple groups.
We review next the structure of . Let be the class of all groups that are either finite or abelian (or both), and let be the class of all ordinals. Let , , and assume we have defined for each , . Each is either a limit ordinal or a successor ordinal. If is a limit ordinal, we set
If is a successor ordinal, let be the class of groups which can be obtained from members of by no more than one operation of extension or directed union.
Theorem 9.2 ().
We have that .
Proof of Theorem 9.1.
Let . For , let H be the following statement:
for , , and , there exists such that every locally finite Cayley graph of admits a harmonic, strong graph height function of the form .
Now, is the set of infinite, finitely generated, abelian groups. By [22, Prop. 4.3, Thm 5.2(b)], any locally finite Cayley graph of has a group height function, and hence a harmonic, strong graph height function of the form . Therefore, H holds, and we turn to the induction step.
Let , , and assume H holds for all . Let with the smallest such ordinal. There are two cases to consider, depending on whether or not is a limit ordinal. If is a limit ordinal, by (9.1), there exists , , such that . The claim now follows by H.
We assume for the remainder of this proof that is a successor ordinal. By Theorem 9.2, the group is obtained from groups in by exactly one operation of either extension or directed union. That is, there are two sub-cases to consider.
There exist such that is isomorphic to a normal subgroup of , and .
There exist a directed set and a family satisfying
Assume (a) holds. Since is finitely generated, so is .
Suppose is infinite. We shall use the fact that . Let be a finite set of generators of with and , and let be the corresponding Cayley graph of . A locally finite Cayley graph of may be constructed as follows. Let
be the (finite) generator set of derived from . The vertex-set of is the set of cosets , and two such vertices , are connected by an edge of if and only if there exist , such that and are connected by an edge in .
By H, there exists , not depending on the choice of , such that admits a harmonic, strong graph height function . Let and be given by
The following lemma completes the proof of this case.
We have that:
, and acts quasi-transitively on by left-multiplication,
the pair is a harmonic, strong graph height function of .
(a) Since , we have that for , whence
It is elementary that, for and ,
Since is a graph height function, we have that . Let be the cosets of in , and let
We show next that each is contained in an orbit of acting on . (Actually the are the orbits.) It follows that acts quasi-transitively on .
Without loss of generality, let . We shall show that there exists such that . Suppose , where . There exists such that , which is to say that .
There exist such that , . Then, , and by (9.2).
(b) It is trivial that . For and , we have
|since is normal|
|since is -difference invariant,|
Therefore, is -difference-invariant.
For , there exist such that
whence, since is a normal subgroup of ,
In conclusion, is a strong graph height function of .
We show finally that is harmonic on the Cayley graph . The edges incident to the vertex labelled have the form for . Since is harmonic on the quotient graph, it suffices to show that the cardinality does not depend on the choice of . For and , if and only if , which is to say that , whence . ∎
Suppose is finite. Since , we have that and . By H, there exists with such that any locally finite Cayley graph of admits a strong graph height function of the form .
Since , there exists (by Poincaré’s Theorem for subgroups) a subgroup that is normal in with finite index, that is, and . Choose a locally finite Cayley graph of , and find a strong graph height function of the form . Let be the restriction of to .
The function is a group height function on the group .
As noted in [22, Remark 4.2], a group height function is a homomorphism from to that is not identically zero. For ,
since and is -difference invariant. Therefore, is a homomorphism.
It suffices now to show that on . Assume the converse, that on . For , there exists such that , so that for . Since is -difference-invariant,
Now , so we may restrict consideration to only finitely many . Therefore, is bounded, which is impossible since is a graph height function. We deduce that on . ∎
Let be a locally finite Cayley graph of . The triple satisfies the conditions of [22, Thm 3.5] with acting by left multiplication, and it follows that possesses a harmonic graph height function of the form .
Assume (b) holds. Let be finitely generated with finite generator set . Since , there exists such that . Let , so that . Then , which contradicts the minimality of . ∎
10. Criteria for the absence of height functions
Let where , and let be the corresponding Cayley graph. Let be the set of permutations of that preserve up to isomorphism, and write for the identity. Thus, acts on by: for with , we have . It follows that . For with , and , we define by , . Write for the subgroup containing all such , and note that operates on in the manner of with left-multiplication.
The stabilizer of is the set of automorphisms of that preserve , that is,
If has finite index, the subgroup satisfies and .
If has a graph height function, then it has a strong graph height function.
(a) Let , and write . Then , which is to say that , and thus so that . Note for future use that
(b) Let be a finite-index normal subgroup, and let . Viewed as automorphisms, we have that , and hence . For , , we have that , since . Therefore, .
Since and , we have that
which implies , as required.
(c) Let be a graph height function of . Since is a finite-index normal subgroup of , by part (b), there exists that is a finite-index normal subgroup of . Since , acts on and is -difference invariant, whence is a strong graph height function. ∎
Let have Cayley graph satisfying .
If has no proper, normal subgroup with finite index, any graph height function of is also a group height function of .
If every element in has finite order, then has no graph height function.
(a) Let be a graph height function of . If satisfies the given condition then, by Proposition 10.1(b), . Therefore, is a graph height function and hence a group height function.
(b) If has a graph height function, by Proposition 10.1(c), has a strong graph height function . Assume every element of has finite order. For with , we have that , whence on .
We now use the argument around (9.5). For , find such that . Since is -difference-invariant, there exists such that
Now , so we may consider only finitely many choices for . Therefore, is bounded on , in contradiction of the assumption that it is a graph height function. ∎
11. Proof of Theorem 5.1
The main step is to show that
where is the identity of . Once this is shown, claim (a) follows from Corollary 10.2(b) and the fact that every element of the Grigorchuk group has finite order, . It therefore suffices for (a) to show (11.1), and to this end we study the structure of the Cayley graph .