SIMPLICIAL APPROXIMATION AND COMPLEXITY GROWTH

# Simplicial approximation and complexity growth

Abstract: This work is motivated by two problems: 1) The approach of manifolds and spaces by triangulations. 2) The complexity growth in sequences of polyhedra. Considering both problems as related, new criteria and methods for approximating smooth manifolds are deduced. When the sequences of polyhedra are obtained by the action of a discrete group or semigroup, further control is given by geometric, topologic and complexity observables. We give a set of relevant examples to illustrate the results, both in infinite and finite dimensions.

## 1 Introduction

Analysis situs, an ancestor of modern topology, arose as a clandestine area of mathematics in the nineteenth century. Gradually it became more accepted, thanks to the work of H. Poincaré, P. Alexandrov, O. Veblen, H. Hopf, J. Alexander, A. Kolmogorov, H. Weyl, L. Brouwer, H. Whitney, W. Hodge and S. Lefschetz, among others.

One of its driving forces, the approximation of shapes (and spaces) through the juxtaposition of prisms or polyhedra, permeated to science and art, becoming essential in our view of the world. From P. Picasso’s cubism to quantum gravity, human perception seemed to accept simplices as elementary blocks to approach forms and space.

It is standard, from a mathematical perspective, to infer estimates of error, complexity, and changes in both complexity and error in a process of approximation, also to estabilish quantitative and qualitative criteria for convergence, and infer bounds for the speed at which such a convergence (if any) occurs.

This paper is motivated by those problems; we obtain, using suitable tools, results of this kind for evolving polyhedra on manifolds. We describe sequences of complexes associated to coverings of spaces by open sets. Sequences of this type, considered by P. Alexandrov (see [Ale-Pon]) under the name of projective spectra, yield, under suitable convergence assumptions, approximations of a paracompact Hausdorff space up to homeomorphism.

We regard the number of simplices and the dimension of each complex in the sequence as a measure of its complexity, and control its growth not only in the limit, but also at every stage. This delivers sequences of irreducible complexes, those for which the excess of complexity is eliminated, say. If the space in question is a differentiable manifold endowed with a Riemannian metric, those irreducible complexes, together with available tools from geometric measure theory, yield a quantitative approximation as well.

To perform those constructions in a systematic way, we consider actions of discrete groups and semigroups, say , on complexes associated to coverings by open sets. We describe -representations/actions that yield convergent sequences of complexes, to make a connection with expansive systems, or e-systems; in those systems the convergent sequence of complexes is obtained by iteration of a suitable initial simplicial complex, a generator, say.

If the space where acts expansively is a closed Riemannian manifold, estimates for the minimal complexity of the generating complex are achieved. This is possible thanks to comparison results in differential geometry.

We briefly mention the contents of this work.

Sections 2.1, 2.2 and 2.3 provide some notation and framework.

In Section 2.4 complexity functions for simplicial complexes are proposed, and we mention their main and useful properties.

Section 2.5 deals with concrete realizations of complexes in Euclidean space; this is needed, together with the functions introduced in Section 2.4, to obtain better approximations of spaces when compared with those achieved by arbitrary convergent sequences (Section 2.6); this is developed in Section 2.8 both from a qualitative and quantitative perspective.

In Section 2.7 increasing sequences of numbers control the complexity growth in sequences of complexes constructed from finer and finer coverings, as measured by the functions introduced in Section 2.4, yielding a quantitative description of the process in the limit. Those growths are measured by what we call the simplicial growth up to dimension , denoted by , and by the dimension growth, denoted by Dim. In fact is a generalization of what is known as topological entropy (see [Wal]), meanwhile Dim is a relative of mean dimension (see [Gro3]).

Section 3 begins with a natural framework for groups and semigroups actions on spaces, usually known as -spaces. We mention the natural morphisms between objects of this type, some advantages of this perspective, to define the evolution of simplicial complexes in -spaces, where the growths of complexity can be measured.

In Section 3.1 the exponential growth of the -simplices is studied under assumptions on , to infer some quantitative control at every stage.

In Section 3.2 we describe a particular type of -spaces, namely -spaces with property-e. The first remarkable issue of the expansive property, or property-e, is that it can be characterized using either topological (set theoretic) or geometric tools. The set theoretic characterization leads to the concept of a generator, an open cover that has a good response to the action of , say. It could be seen as a complex that under the action of evolves towards an acceptable approximation of the space. We describe in which sense the evolving nerves of generating covers approximate the space, and recall fundamental results in geometry and topology that suit our developments. All the results from previous Sections can be used in this scenario, and the adaptation of them is left to the reader.

In Section 3.3, assuming that the space is of Riemannian type, we provide estimates to have a better control of the generating process. Those estimates find concrete applications in Section 5.

Finally, in Section 4 and Section 5 we present some examples. Section 4 deals with infinite dimensional examples where estimates for the simplicial growth, as measured by the family , and the dimension growth, as measured by Dim, appear. Section 5 describes finite dimensional closed manifolds for which an expansive action can be constructed. Some of the examples in finite dimension are not new, and the list of examples is far from being exhaustive nor definitive; their (not so detailed) description is included for many purposes:

1. To ensure that the results of Sections 3.2 and 3.3 are non-void, enabling concrete constructions and estimates.

2. To have an idea of the methods used to construct them.

3. To allow the construction of new examples from known ones.

Sections 4 and 5 are not entirely independent: all the examples in Section 5 can be used in Section 4.1 to construct infinite dimensional closed manifolds with property-e.

## 2 Simplicial complexes, complexity and convergence

We state properties of the canonical simplicial complex associated to the covering of a space by open sets, known as the nerve of the covering. Some statements can be found in [Ale-Pon], [Hur-Wall], [Lef], and the references therein. Other properties are new (at least for the author), and all of them will be used in this article.

### 2.1 The nerve of open covers

If is a compact Hausdorff space1 we denote by the set of covers of by open sets: one calls the members of open covers.

###### Remark 2.1.

Since is compact, it suffices to identify with the totality of all finite covers by open sets of to simplify.

If and belong to , one says that is finer than if whenever is an element in there exists some in such that , and writes if that is the case. This notion induces a partial order on .

If one denotes by the refinement of by (or equivalently the refinement of by ): its elements are intersections of one element from and another from . One can write

 α∩β:=l.u.b.{ γ | γ≻α,γ≻β },

where l.u.b. denotes the supremum (or least upper bound) in induced by .

###### Remark 2.2.

One can play further with those notions and use the language of lattices, something that we give for granted.

Let be given as , where is an indexing set (finite since is compact). Associated to is a simplicial complex, known as the nerve of , that we denote by , uniquely defined up to homotopy, and whose simplices are constructed as follows: for every in the set of -dimensional simplices of , denoted by , is given by

 { [ai(0),...,ai(k)] | k⋂r=0Ai(r)≠∅ },

where for each in we identify the open set with the -simplex .

### 2.2 Dimension, simplicial mappings and irreducibility

Given in , for every we denote by the cardinality of , i.e. the number of -simplices in . By those means one introduces the dimension of , denoted by , as the maximal for which is different from zero.

For and in with there exists a simplicial map from to , say , defined up to homotopy, satisfying the following properties:

1. If then .

2. Whenever and is in , then the image of under is completely determined by the image of the -simplices making up : this allows the possibility that is in for some (for example when different vertices of are mapped to the same -simplex in ).

One says that is compatible with . It is important to note:

1. Such a map need not be unique.

2. If and we have constructed two simplicial maps and compatible with , then we have a simplicial map given by that is also compatible with .

There are open covers we distinguish for later purposes.

###### Definition 2.3.

One says that in is irreducible if no open refinement of has a nerve isomorphic with a proper sub-complex of , i.e. if there is no finer than that admits a strict simplicial embedding from its nerve to the nerve of .

###### Lemma 2.4.

Irreducible covers have the following properties:

1. If is irreducible then every member of it contains a point in that is not contained in other member.

2. If is irreducible then whenever all the simplicial maps from to compatible with are surjective.

3. If is compact, then every in has an irreducible refinement (one says that irreducible covers are cofinal in .

4. If is a manifold whose (real) dimension is equal to , then for every irreducible in one has .

###### Proof.

1: If has some element, say , that contains no point that is not contained in the rest of the ’s, then is a refinement of and is a proper subcomplex of .

2: This is clear from the definition.

3: Let in be given, and consider a sequence in so that , where all the ’s are reducible, and such that the corresponding nerves form a sequence of complexes, with being a proper subcomplex of where . By compacity of the sequence must stop, and the last term is an irreducible refinement of .

4: Being a manifold of dimension , it suffices to prove the result on some open set homeomorphic to , where the statement is obviously true. ∎

### 2.3 Chain complexes, homology

To handle a better notation, given an open cover , we denote by the set of injective mappings such that , modulo permutations. By those means we identify the -simplex with whenever is in . Therefore we have a bijection between and .

If is an Abelian group one identifies with the (Abelian) group of -chains in with coefficients in , so that

 Ck(α,G):={ ∑→i∈∧k(α)g→i σk→i | g→i∈G }.
###### Remark 2.5.

Given a permutation of letters, say , we are identifying with in , although in we have , where denotes the sign of .

Introduce the boundary operator, denoted by , as the map that sends -chains to -chains in a -linear way. Since is generated by the elements in , it suffices to define the action of on the elements of .

Thus given in we set

 ∂σk→i:=k∑r=0 (−1)r σk−1→i∖i(r),

where provided that , where means that is deleted.

One verifies that for every in one has in i.e. the boundary of the boundary of every -chain is equal to zero.

Using the boundary operator one defines two subgroups of :

1. The subgroup of -cycles, denoted by , and defined through

 Zk(α,G):=Ck(α,G) ∩{ c | ∂c=0 }.
2. The subgroup of -boundaries, denoted by , and defined through

 Bk(α,G):=Ck(α,G) ∩ ∂Ck+1(α,G).

By those means the -th homology group of with coefficients in is defined, namely

 Hk(α,G):=Zk(α,G)Bk(α,G).
###### Remark 2.6.

An algebraist would say that measures the inexactness of the sequence

 ....→Ck+1\lx@stackrel∂→Ck\lx@stackrel∂→Ck−1→...

A geometer/topologist would say that measures the amount of closed -chains that are not filled in the space in question (i.e. that are not boundaries) up to bordism.

Let be the graded -module associated to the homology of with coefficients in In particular if is taken as , one denotes by the -th Betti number of . Regarding the structure of the complex , one has the isomorphism .

If no confussion arises we identify and with the real dimension of and respectively, whence in particular and follow.

Using the previous nomenclature one defines , so that is the Euler-Poincaré characteristic of . The equalities for and entail that is equal to the sum .

From the definitions/constructions one has the equality for every in , therefore:

###### Lemma 2.7.

Whenever is in one has the identity

 χ−1(α)=dimK(α)∑i=0(−1)i|△i(α)| .

### 2.4 Complexity functions

In this Section we define complexity functions for the simplices of an open cover on We infer some properties of their minimizers and some estimates for them. The next observation is fundamental.

###### Lemma 2.8.

For every in and in the minimum of among those ’s finer than is obtained for irreducible ’s. In particular, if is irreducible, then the minimum mentioned above is obtained for itself. The same is true for the sum and for the dimension .

###### Proof.

Follows from 3 and 2 in Lemma 2.4, namely that irreducible covers are cofinal in the directed set : hence if is irreducible then for every finer than all the simplicial maps from the nerve of to the nerve of compatible with are surjective. ∎

To quantify the complexity of , that we measure in terms of its dimension and its number of simplices, also by similar quantities in whenever is finer than , we introduce the functions and from to through:

 DimK(α):=minβ≻αdimK(β),
 Gk(α):= k∑i=0|△i(α)|

and

 Sk(α):=minβ≻αGk(β).

From the definitions, Lemma 2.8, and the identity in Lemma 2.7 we observe:

###### Lemma 2.9.

For every in we have

1. If is larger than zero

 G0(α) ≤ Gk−1(α) ≤ Gk(α) ,
 maxl∈{0,...,k} |△l(α)| ≤ Gk(α) ≤ (k+1)maxl∈{0,...,k} |△l(α)|,

with

 |△l(α)| ≤ (G0(α)l+1) ≤ G0(α)l+1(l+1)!.
2. is equal to for some irreducible finer than .

3. is equal to for some irreducible finer than .

4. The identity

 χ−1(α)=2 dimK(α)−1∑i=0(−1)i Gi(α)+(−1)dimK(α) GdimK(α)(α).

### 2.5 Euclidean realization of nerves

Let be an open cover for . We say that a partition of unity for is compatible with if it satisfies the following conditions:

1. for every in .

2. For every in we have that whenever is not in .

Identify the -simplex of corresponding to with the unit vector in along the -th direction, to denote the image of the map

 x:V⟶R|I|
 v↦x(v)=∑i∈I xi(v)[ai]

by , and call it an Euclidean realization of . Observe that different partitions of unit on compatible with induce maps from to that are homotopic.

Sometimes we identify with a polyhedral current in . We do this as follows: for every in we have the -current that corresponds to the pure point measure supported at distance one from the origin along the -th axis. Using the convention of Section 2.3, for in we identify with the polyhedral -current , where is the -dimensional Hausdorff measure on whose support is the convex hull of , meanwhile is a -vectorfield of unit length tangent to such a plane (see [Fed] for all the details).

Observe that the chain complex associated to is isomorphic with that defined in Section 2.3 for .

### 2.6 Sequences of nerves: convergence

The results in this Section are a simplified version, suitable for the applications in this work, of general results attributed to P. Alexandrov, S. Lefschetz and V. Ponomarev (see [Ale-Pon]-[Lef]). In the literature the nomenclature is not uniform: we try to unify some notions as well.

Consider a sequence in with . If is identified with the simplicial complex that corresponds to the nerve of , then for every we have a simplicial map compatible with and defined up to homotopy (see Section 2.2). Those simplicial maps can be composed inductively to get a map from to whenever in the usual way, where .

We have an infinite sequence of simplicial complexes and mappings making up a directed set .

One says that the sequence is convergent if every member of consists at most of a point when goes to infinity.

As tends to infinity we have a surjective simplicial map from to for every . We naturally identify the inverse or projective limit of the directed set with the nerve of when tends to infinity, that we denote by

 lim←(K(αn),Tn),

to state:

###### Proposition 2.10.

(Alexandrov-Ponomarev [Ale-Pon]) Assume that the sequence

 (K(αn),Tn)n∈N

is convergent. Then when goes to infinity the nerve of and are homeomorphic.

###### Proof.

We describe the projective limit of the directed set to see that there is a homeomorphism between such a limit and .

Let be a sequence of simplices, with in for every We say that is an admissible sequence or a thread for if whenever is larger than , and say that an admissible sequence is an extension of if for every the simplex is a face (not necessarily a proper one) of . If the admissible sequence has no extensions other than itself, we say that it is a maximal admissible sequence (or a maximal thread).

Provide with the following topology: given a simplex in for some , a basic open set around consists of all maximal admissible sequences such that is a face of . In such a way one generates a topology for the limit space, namely the set of all maximal admissible sequences.

Whenever is a point in we have a simplex in that corresponds to all the open sets in to which belongs; due to the convergence assumption we note that is a maximal admissible sequence, and conversely, every maximal admissible sequence in is of the form for some in .

Therefore is isomorphic to the inverse limit of , and at this stage it is easy to see that they are homeomorphic. ∎

###### Remark 2.11.

Neither a metric nor a differentiable structure on are required in Proposition 2.10.

###### Remark 2.12.

Proposition 2.10 can be refined sometimes: it might happen that for some finite all the elements in together with their intersections are contractible (see Figure 1 in Section 5). Then is said to be a ‘good cover’, and it is known that in such a case is homotopically equivalent to (see [Hat] for example).

One is led to consider convergent sequences of coverings to reconstruct and/or approximate a given space up to homeomorphism in the limit. On every paracompact Hausdorff space a convergent sequence can be constructed in an arbitrary way. It is of interest, however, to create them under some quantitative and qualitative control. We will see in Section 2.7 that the family of complexity functions introduced in Section 2.4 are of much use for those purposes. Moreover, if we endow with a Riemannian metric, one can consider subsequences of complexes associated to those complexity functions, and have a better approximation of in the limit (Section 2.8).

### 2.7 Controlling sequences

Let be a sequence of nerves and simplicial mappings built up from a sequence of open covers for , with . From Lemma 2.9 we know that if we consider the sequence of positive integers there exists, for every in , at least one irreducible finer than so that is equal to .

Fix and let be a sequence of irreducible covers that achieve, for each in , the minimum of . Since is finer than , then when goes to infinity we have, under the hypothesis of Proposition 2.10, that is homeomorphic to ; since is irreducible the dimension of is equal to the dimension of .

For every consider the increasing sequence of positive integers ; each sequence goes to infinity as increases. The next Proposition provides a correlation between those sequences thanks to Lemma 2.9.

###### Proposition 2.13.

Let be a compact Hausdorff space, without a boundary, whose topological dimension is uniform and finite. For a fixed let be a sequence of irreducible open covers associated to a convergent sequence . Then as goes to infinity we have the equality

 χ−1(V)=2 dimV−1∑i=0(−1)i Gi(βk,n)+(−1)dimV GdimV(βk,n).

To have more control on a sequence we consider strictly increasing sequences of positive real numbers, say , going to infinity and such that

 0

If the sequence satisfies those estimates, we say that it controls the simplicial growth of up to dimension . If exists, then

 ent↓k(αn,c(n))=ent↑k(αn,c(n))=:entk(αn,c(n)).

Similarly, if

 0

we say that controls the dimension growth of .

Of course if exists, then

 Dim↓(αn,c(n))=Dim↑(αn,c(n))=:Dim(αn,c(n)).

Using Lemma 2.9 we deduce:

###### Theorem 1.

Assume that controls the simplicial growth of up to dimension for some finite . Then is a controlling sequence for the growth of simplices of up to dimension for every finite .

###### Proof.

From Lemma 2.9 we see that

 G0(αn)≤Gk(αn)≤(k+1) maxl∈{0,...,k} G0(αn)l+1(l+1)!,

therefore we have

 ent↓0(αn,c(n))≤ent↓k(αn,c(n))≤(k+1)ent↓0(αn,c(n)),

and similarly

 ent↑0(αn,c(n))≤ent↑k(αn,c(n))≤(k+1)ent↑0(αn,c(n)).

By a simple interpolation we deduce that and are comparable if both and are finite, and similarly for and . ∎

Natural choices for controlling sequences are:

1. , and then (in the strict sense)

1. The simplicial growth is of exponential type.

2. The dimension growth is of linear type.

2. , and then

1. The simplicial growth is of polynomial type.

2. The dimension growth is of logarithmic type.

Of course there are other possibilities.

Theorem 1 says that the order of the simplicial growth up to dimension in a sequence is comparable to the order of growth of -simplices if is finite or is finite dimensional.

On the other hand, observe that a necessary condition for to have a sequence controlling its dimension growth is that the underlying space must have infinite topological dimension, i.e. the supremum of as varies in must be unbounded; this condition is also sufficient if is convergent.

###### Remark 2.14.

We understand that is the standard sequence; due to that we will omit from the expressions whenever such a sequence is used.

### 2.8 Life with a Riemannian metric

Now is a smooth closed manifold, provided with a distance function arising from some Riemannian metric , say . Then the convergence of is equivalent to the statement that all the members of have a diameter that goes to zero as goes to infinity, where we assume that for every .

As in Section 2.7 consider, for every and , an irreducible cover finer than so that is equal to . Then for each we have a simplicial embedding from the nerve of to the nerve of , but there is no guarantee that , nor that the members of are contractible.

But we can do better; since the diameter of the members of are decreasing as increases, then for every there exists some large enough such that every member of the irreducible cover has a diameter smaller than some Lebesgue number of , yielding a surjective simplicial map .

Assume that is fixed: we have a subsequence of made up of irreducible covers and endowed with surjective simplicial maps making up a directed set . We recover Proposition 2.10 for the projective limit

 lim←(K(βϕ(n)),Tϕ(n)),

although with better quantitative control. This is due to the results in Section 2.7, and because the dimension of is bounded by the dimension of at every stage (Lemma 2.4).

If is the largest diameter of a member in , we assume that is large enough so that is smaller than the injectivity radius of . Then for every in we can choose some in such that whenever , and identify with the -simplex that corresponds to .

Let be an Euclidean realization of (Section 2.5). Embed the -simplices of in using a Lipschitz map so that the image of the -simplex corresponds to the distance minimizing path or geodesic between the points and , that we regard as a rectifiable path (or current) in . We observe (see [Gro2]):

###### Lemma 2.15.

Endow the set of -simplices in , namely , with the distance induced by the embedding of the -simplices in , extending it to all in the natural way; denote such a distance by . Then, as goes to infinity, the metric space converges to in the Gromov-Hausdorff sense.

Consider , the graded -module of general currents on , and for every in let denote the graded -module of polyhedral currents on the Euclidean realization of . Hence if is of Lipschitz type, then we get a linear map .2

Let be the mass norm on induced by the Riemannian metric . A fundamental fact, to be found in [Fed], asserts that the closure in with respect to of pushforwarded polyhedral currents by Lipschitz maps into is , the -module of rectifiable currents on . An important sub-module of , denoted by , is the -module of integral currents; it consists of rectifiable currents whose boundary is also rectifiable. The choice of an atlas on and elementary constructions from geometric measure theory give:

###### Proposition 2.16.

and depend on the Lipschitz structure on chosen.

Embed now the -simplices of in by means of a Lipschitz map using the geodesics that correspond to the -simplices as their boundary, straightening them as much as possible, so that in the process their mass (with respect to ) tends to minimize.

Then proceed inductively to get, for every and each not bigger than the dimension of , a Lipschitz embedding of the -simplices in the Euclidean realization of inside , all whose images have a diameter not larger than ; we obtain a map at the level of currents

 yϕ(n)♯:P∗(|K(βϕ(n))|)↪I∗(V)⊂R∗(V).

More precisely3, if is in , then is almost minimal among those rectifiable currents whose boundary is , for every . By almost minimal we mean that the minimum of mass might not occur, however there is a sequence of Lipschitz maps leading to an infimum.

This process of approximation gives:

###### Theorem 2.

Let be a smooth closed Riemannian manifold. Let be a sequence in , with , and such that the diameter of each member of goes to zero as goes to infinity. Then for every and every positive there exists some , a cover minimizing , and a Lipschitz map such that

 Mg( ym♯|K(βk,m)|−V )<ϵ,

where is the current representing , and is the polyhedral current that corresponds to an Euclidean realization of . If the dimension of is not , such a map is independent of the Lipschitz structure on .

###### Remark 2.17.

We know, thanks to the work of E. Moise, S. Donaldson and D. Sullivan, that only in dimension there are smooth manifolds that are homeomorphic but not Lipschitz equivalent.

## 3 The category of Γ-spaces

We denote by a countable or discrete group or semigroup whose cardinality is . Let be a representation of on the set of mappings of , where we understand, if is a group, that is a sub-group of , the group of homeomorphisms on . If is a semigroup, then is a sub-semigroup of , the semigroup of endomorphisms on . We denote a structure of this type by a tuple , and speak of a system, a representation of , or a -space.

###### Remark 3.1.

We restrict to discrete groups and semigroups since:

1. Some of the results are not valid for arbitrary ’s.

2. All the examples in this article belong to this class.

We identify the totality of those structures with the objects in the category of -spaces. The standard morphisms between objects in this category are:

1- Conjugations: Two systems and are said to be conjugated if there exists a homeomorphism that interwinds the action of , i.e. a homeomorphism between and that is -equivariant. The notion of being conjugated is a strong equivalence relation; not only the underlying (topological-geometric) spaces in question are homeomorphic, moreover the dynamics induced by the maps are, up to a continuous change of coordinates say, equivalent.

2- Factors/Extensions: is said to be an extension of , or is said to be a factor of , if there is a continuous surjection , so that whenever is in we have .

###### Remark 3.2.

Of course if is both a factor and an extension of , then both systems are conjugated.

###### Remark 3.3.

The advantage of using the language of categories in this context is that some operations of algebraic topology (for example the loop functor, the suspension functor and the smash product) can be used as self-functors. By those means one obtains new systems from known ones (see Section 5.3.3).

Consider the action of ’s inverses on elements of . If is in we have a map and also an induced map where in the case of semigroups we understand that is given by for every subset of . Whenever is a finite subset of and is in we set

 αF:=⋂γ∈Fρ(γ)−1α.

In what follows we describe as increases both from a quantitative and a qualitative perspective.

### 3.1 Exponential simplicial growth: topological entropy

Assume that is generated by a finite subset of elements, say , where we assume, if is a group, that contains all its inverses, i.e. that . Then whenever is a finite subset of we define its boundary with respect to , that we denote by , as the subset of made up of those ’s such that is not in for some in .

Consider an increasing sequence of subsets exhausting , say . Such a sequence is said to be of Følner type if the quotient between and goes to zero as goes to infinty. If such a sequence exists, then is said to be amenable (see [Gro2] for more about this).

###### Remark 3.4.

Since all the results in Section 2.7 are valid in this context, we will not repeat analogous statements unless this is relevant; one should replace by , and by

 ⋂γ∈F(n)ρ(γ)−