On the Residual Finiteness Growths of Particular Hyperbolic Manifold Groups

# On the Residual Finiteness Growths of Particular Hyperbolic Manifold Groups

## Abstract.

We give a quantification of residual finiteness for the fundamental groups of hyperbolic manifolds that admit a totally geodesic immersion to a compact, right-angled Coxeter orbifold of dimension 3 or 4. Specifically, we give explicit upper bounds on residual finiteness that are linear in terms of geodesic length. We then extend the linear upper bounds to hyperbolic manifolds with a finite cover that admits such an immersion. Since the quantifications are given in terms of geodesic length, we define the geodesic residual finiteness growth and show that this growth is equivalent to the usual residual finiteness growth defined in terms of word length. This equivalence implies that our results recover the quantification of residual finiteness from [BHP] for hyperbolic manifolds that virtually immerse into a compact reflection orbifold.

###### Key words and phrases:
Residual finiteness growth, hyperbolic manifolds, right-angled Coexter group
###### 2010 Mathematics Subject Classification:
Primary: 20E26; Secondary: 57M10, 20F65

## 1. Introduction

It is well-known that separability properties on groups have deep connections with basic problems in group theory. In the 1940’s, Mal’cev demonstrated that separability properties like residual finiteness and conjugacy separability produce solutions to the word and conjugacy problems, respectively, for finitely presented groups [Malcev]. In more recent years, separability properties have also played a fundamental role in low dimensional topology. In [Scott], P. Scott gave an important topological reformulation of subgroup separability when the groups in consideration are the fundamental groups of manifolds; separability allows one to promote an immersed compact set to an embedded one in a finite cover. This topological reformulation of subgroup separability, usually called Scott’s criterion, played a crucial role in the recent resolutions of Walhausen’s Virtually Haken Conjecture and Thurston’s Virtually Fibered Conjecture (see [AgolGrovesManning], [Wise:QCH], [Haglund]).

The simplest separability property, residual finiteness, allows us to separate nontrivial group elements from the identity using finite index subgroups. More precisely, a group is residually finite if for every non-identity element , there exists a finite index subgroup of such that . Quantifying residual finiteness, a concept first introduced by K. Bou-Rabee in [Bou1], refers to bounding the indexes of the finite index subgroups in terms of algebraic data about . In studying separability properties of the fundamental groups of manifolds, the bounds can also be given in terms of geometric data about the manifolds as in [PP12]. A quantification of residual finiteness informs us on the minimal possible index of a subgroup of that separates from the identity, and it has been studied for various classes of groups including free groups, surface groups, and virtually special groups (see for instance [Bou1], [BHP], [Bou2], [BM2], [Buskin], [Kass], [KozmaThom], [PP12], [Rivin]).

The reasons for quantifying residual finiteness and studying residual finiteness growths are varied. First, residual growths can help distinguish classes of groups. For example, in [BM14] Bou-Rabee and McReynolds give a characterization of when non-elementary hyperbolic groups are linear in terms of residual growths. Additionally, a quantification of residual finiteness can serve as a foundation on which to build an approach to the quantification of stronger separability properties. In [PP12], the author presented a quantification of the residual finiteness of hyperbolic surface groups in terms of geodesic length. The result was then used to make effective a theorem of P. Scott [Scott] that these groups are also subgroup separable. The author used a key insight of Scott that all hyperbolic surface groups arise as subgroups of a particular right-angled Coxeter group, generated by reflections in the sides of a regular, right-angled pentagon in . Quantification proofs also usually proceed algorithmically and can provide insight into how to construct the finite index subgroups/covers associated to residual finiteness and subgroup separability.

To keep with the notation present in most of the literature on quantifying residual finiteness we introduce the residual finiteness growth, originally defined in [Bou1] as follows. For a group with a fixed finite generating set , let the divisibility function be defined by

 DG(g)=min{[G:H]:g∉H and H≤G}.

When is residually finite, of course takes values in . We note that hyperbolic manifold groups are finitely generated linear groups and are thus known to be residually finite by the work of Mal’cev [Malcev]. Define the residual finiteness function to be the maximum value of on the set

 {g∈G−{1}:∥g∥S≤n},

where is the word-length norm with respect to . The growth of is called the residual finiteness growth.

In this paper, we are concerned with the residual finiteness growths of the fundamental groups of particular hyperbolic manifolds, and we give all quantifications of residual finiteness in terms of geometric data about the manifolds. Accordingly, we define a new function for a hyperbolic manifold by letting be the maximum value of on the set

 {α∈π1(M)−{1}:ℓρ(α)≤n},

where is the length of the unique geodesic representative of with respect to the hyperbolic metric on . The growth of is called the geodesic residual finiteness growth.

In order to state our results in the language of growth functions, we introduce the following standard notation: For functions , we write if there exists such that . Further, we write if and .

When is a compact hyperbolic manifold, the residual finiteness growth and the geodesic residual finiteness growth are the same, i.e. , due to an application of the Svarc-Milnor Lemma (see [BridsonHaefliger, P. 140]). However, we believe that the relationship between the two growths for non-compact, finite volume hyperbolic manifolds might be more complicated. We give a proof of the equality of the growths in the compact case (see Lemma 6.1) and a brief justification of the reasoning for the non-compact case in Section 6.

This paper studies the residual finiteness of the fundamental groups of hyperbolic manifolds that admit a totally geodesic immersion to a compact, right-angled Coxeter orbifold of dimension 3 or 4. Begin by letting be any compact polyhedron in (resp. ), all of whose dihedral angles are , which we will refer to as a compact all right polyhedron. These polyhedra serve as the analog of Scott’s right-angled pentagon in [Scott]. (Note that compact all right polyhedra only exist in dimension up to 4 by the work of Vinberg [Vinberg]). We denote by the right-angled Coxeter group of isometries of (resp. ) generated by reflections in the codimension–1 faces of . The quotient (resp. ) is the compact, right-angled Coxeter orbifold, , defined by . The orbifold is also often called a compact reflection orbifold in the literature.

Drawing from the work of Agol, Long, and Reid in [ALR], and making use of an observation of Agol in [Agol], we obtain the following theorems, which constitute the main results of the paper and which generalize [PP12, Theorem 6.1]:

###### Theorem 3.3.

Let be a hyperbolic manifold that admits a totally geodesic immersion to a compact, right-angled Coxeter orbifold, , of dimension 3. Then for any , there exists a subgroup of such that , and the index of is bounded above by

 2πVPsinh2(ln(√3+√2)+dP)ℓρ(α),

where is the length of the unique geodesic representative of , and where and are the diameter and volume of , respectively.

In Section 4, we establish a 4–dimensional analog of Theorem 3.3:

###### Theorem 4.3.

Let be a hyperbolic manifold that admits a totally geodesic immersion to a compact, right-angled Coxeter orbifold, , of dimension 4.Then for any , there exists a subgroup of such that , and the index of is bounded above by

 8π3VPsinh3(ln(2+√3)+dP)ℓρ(α),

where is the length of the unique geodesic representative of , and where and are the diameter and volume of , respectively.

In the notation defined above, we then have the following corollary:

###### Corollary 4.4.

Let be a hyperbolic manifold admitting a totally geodesic immersion to a compact, right-angled Coxeter orbifold of dimension 3 or 4. Then, the geodesic residual finiteness growth is at most linear. That is to say, .

###### Remark 1.1.

We note that in the above theorems and corollary, the hyperbolic manifold need not be compact. In dimension 2, the compactness criteria of [PP12, Theorem 5.4, Theorem 6.1, and Theorem 7.1] is also not necessary. Indeed, every hyperbolic infinite area surface of finite type can be tiled by regular, right-angled pentagons and all proofs follow as in the compact case.

###### Remark 1.2.

We also note that the dimension of the hyperbolic manifold need not match the dimension of the orbifold to which it admits a totally geodesic immersion. This is important since [ALR, Lemma 4.6] shows that there exist infinitely many compact, arithmetic hyperbolic 3–manifolds that admit a totally geodesic immersion to a particular compact reflection orbifold of dimension 4 (coming from the 120-cell in ), but do not admit such an immersion to any reflection orbifold of dimension 3.

The next lemma allows us to extend Corollary 4.4 to hyperbolic manifold groups that virtually admit the desired type of immersion. A group (resp. topological space ) virtually has property “” if there exists a finite index subgroup of (resp. finite sheeted cover of ) with property “″.

###### Lemma 5.1.

Let be a hyperbolic manifold and let be a finite index subgroup with . Let be the cover of of degree corresponding to the subgroup . Then the geodesic residual finiteness function for is bounded by that of . That is to say, and hence .

Combining Theorems 3.3 and 4.3 with Lemma 5.1 gives:

###### Corollary 5.2.

If is a hyperbolic manifold that virtually admits a totally geodesic immersion to a compact, right-angled Coxeter orbifold of dimension 3 or 4, then .

We note that in [BHP], K. Bou-Rabee, M.F. Hagen, and the author give a quantification of residual finiteness for right-angled Artin groups (raAgs) in terms of word length. In particular, we prove that the residual finiteness growth of raAgs is at most linear. As suggested by Lemma 5.1, residual finiteness quantifications are essentially preserved under passing to subgroups and finite index extensions. Thus, the quantification for raAgs results in a quantification of residual finiteness for all groups that virtually embed in raAgs, which are called virtually special groups. This class of groups includes Coxeter groups [HaglundWiseCoxeter], and in particular contains the class of manifold groups satisfying the hypothesis of Corollary 5.2. By an application of Lemma 6.1 below, the results of [BHP], therefore, imply Corollary 5.2. However, the proofs of Theorems 3.3 and 4.3 do not rely on the canonical completion and retraction methods of [haglundwise] for cube complexes. Instead, the proofs use simple calculations in hyperbolic space inspired by work of Agol–Long–Reid [ALR] and Agol [Agol]. Most importantly, the calculations lead to the explicit bounds of Theorem 3.3 and 4.3, and obtaining such bounds using [BHP] should require a significant amount of work.

Acknowledgements. Much of the work in this paper originally appeared in the author’s thesis, written under the supervision of Feng Luo, whom the author thanks for his help, support, and encouragement. The author would also like to sincerely thank Ian Agol for sharing his work and ideas, as well as for insightful conversations. Many thanks are also due to Tian Yang for his suggestions regarding the proofs of Lemmas 3.1 and 4.1, to Alan Reid and Nicholas Miller for helpful conversations, to Ben McReynolds for his encouragement to write this paper, and to David Duncan for comments on an early draft. The author would especially like to thank an anonymous referee whose extensive and thorough suggestions have greatly increased the quality of this paper.

## 2. Preliminaries

### 2.1. General notation:

Throughout the paper we switch freely between the Poincaré ball () and half-space () models of hyperbolic space. In the 3–dimensional Poincaré ball model , we define the three hyperplanes , , and . In the 3–dimensional half-space model we define the hyperplanes and . Similarly, in we define the four hyperplanes for as the hyperplanes obtained by restricting to be . The hyperplanes for in are defined by restricting the coordinate in to be .

### 2.2. Topological view of residual finiteness:

When G is the fundamental group of a hyperbolic manifold, we have the following topological formulation of residual finiteness, which is used to prove Theorems 3.3 and 4.3: a hyperbolic manifold group is residually finite if for every there exists a finite index cover of where the unique geodesic representative of does not lift (i.e. no component of the preimage of in projects injectively to ).

### 2.3. Polyhedral convexification:

If is an all right polyhedron in , then by the Poincaré Polyhedron Theorem, the images of under the action of will tessellate .

###### Definition 2.1.

The –convexification of a connected set in , denoted by , is the smallest, convex union of polyhedra in the tessellation of determined by that contains .

Equivalently, we can define the convexification as the intersection of all half spaces bounded by the geodesic hyperplanes in our tessellation of determined by that contain .

## 3. The Proof of Theorem 3.3

Let be a hyperbolic manifold that admits a totally geodesic immersion, , to a compact, right-angled Coxeter orbifold, , of dimension 3. Then the induced map is injective, and can be identified with a subgroup of . When convenient, we blur the distinction between and its image in and simply write . Additionally, the length of the geodesic representative of will be equal to the translation length of .

For with , the cyclic subgroup of generated by will be denoted by . We let be the unique simple closed geodesic of .

The preimage of under the covering map is the geodesic axis, , in for the element . We note that is invariant under the action of on , and therefore, so is , the -convexification of . The image of in is denoted by . Thus, is the smallest closed, connected, convex union of polyhedra in the tessellation of (determined by ) containing . We refer to as the convexification of in . The following lemma, along with Lemma 4.2 in the next section, were first proposed by Agol via private communication [Agol].

###### Lemma 3.1.

Let be the union of polyhedra forming the convexification of   in via the procedure mentioned above. Then any polyhedron must intersect where .

###### Proof.

We define to be the preimage of in . Thus, forms an –neighborhood around the geodesic axis in . Suppose is a polyhedron in our tessellation of which does not intersect . We aim to show that a hyperplane in containing one of the codimension–1 faces of must separate from , demonstrating that and proving the lemma.

We let be the minimal-dimensional face of containing its closest point to and set codim. Note that is also the number of codimension–1 faces of that intersect . Take a shortest geodesic from to that intersects at a point and at a point . Then is an orthogeodesic with since does not intersect the –neighborhood of . We let be a hyperplane through that is perpendicular to , which separates from . We note that necessarily contains .

Case : We first consider the case where is a vertex of and . We begin by sending to the origin of via isometries. Since is an all right polyhedron, the three codimension–1 faces of that intersect necessarily lie in three hyperplanes , and that are the isometric image of , and , which were defined in Section 2.1. Letting , we see that the connected components of are right-angled spherical triangles in with edge lengths .

We note that we can assume that passes through the north pole of the hyperplane (this will be important for our calculation of ). We apply isometries to until , and therefore , lies on the line formed by and lies in . The distance and the radius, , of are inversely proportional. Indeed, Figures (a)a and (b)b show that as the distance between and (the length of ) grows, the radius of becomes small.

We are therefore interested in the threshold of the distance between and so that can be inscribed in a right-angled spherical triangle formed by the boundaries of three pairwise orthogonal hyperplanes. Then if , at least one of the three circles cannot intersect , and thus, one of the three hyperplanes must separate from .

We begin by calculating the radius of a circle inscribed in such a spherical triangle, whose edge lengths are . This calculation follows from an application of classical formulas (see e.g. [Todhunter, Section 89]), but we also include it here. In Figure (a)a, , and are the midpoints of the three edges in our spherical triangle, which are also the points of tangency for the inscribed circle. Since the triangle is formed by the intersection of three pairwise orthogonal hyperplanes in with , the unit sphere in , the length of the edge is equal to , and the length of is .

To calculate the radius of the inscribed circle we apply the following Spherical Law of Cosines. Let be a spherical triangle with angles , and , and with edges of lengths , and opposite the angles , and , respectively. Then,

 cosα=−cosβcosγ+sinβsinγcosa.

For the triangle in Figure (b)b, we have , and therefore, .

Now we calculate the distance length of so that the radius of is . Consider the cross sectional view formed by the intersection of the hyperplane with our setup in Figures (a)a, (b)b above. This view is represented in Figure 3.

Since is a unit circle we know that . Note that the points form a right triangle and we denote the length of the line segment from to by . Therefore, and . Lastly, and , which by a simple calculation gives us .

Case : The case where is an edge of can be handled in a similar way. In this case, we show that .

We assume the same setup as in the previous case where the point on the edge that is closest to is at the origin of and lying on . The extensions of the two codimension–1 faces of that intersect form a pair of orthogonal hyperplanes, and , in . Their boundaries, and , form a spherical bi-disk with angles . We are looking for the threshold such that , and thus , is tangent to such a spherical bi-disk at the endpoints of . Then, if , one of the cannot intersect , and one of the hyperplanes must therefore separate from .

A cross sectional view of this situation is shown in Figure 4 below. Given the triangle in the figure, we know that so that . Thus, .

Case : Lastly, we consider the case where is a codimension–1 face of , that is the case where . Then is an orthogeodesic between the hyperplane containing , which we also call for notational simplicity, and the hyperplane . Taking any is sufficient in this case since itself is a codimension–1 face of whose hyperplane extension separates from (see Figure 5 below).

We take to be the largest of the values in the three cases, i.e. . If does not intersect , then there is a codimension–1 face of whose hyperplane extension separates from so that . Therefore, any polyhedron in that is in the convexification of must intersect the –neighborhood of .

Given the above lemma, we have that , where is the diameter of . The following lemma allows us to calculate the volume of . Again, this calculation is standard (e.g. it is stated without proof in [Meyerhoff]), but it is included here to set the stage for the proof of Theorem 3.3.

###### Lemma 3.2.

We let be the solid tubular –neighborhood of the geodesic segment in between the points and as shown in Figure 6. Then Vol, where is the length of the geodesic between the points and in .

###### Proof.

For this volume calculation we find it convenient to use spherical coordinates. The volume form on , , becomes . Let be the angle in from the positive -axis to the Euclidean ray consisting of points at hyperbolic distance from the -axis, so that the range of values for in is then . Therefore,

,

with the last equality coming from the angle of parallelism laws in hyperbolic space (see [Beardon, Section 7.9]).

We take a subset of the preimage of in that is isometric to and which forms a region like from the previous lemma, where . Lemma 3.2 then implies that Vol Vol. Thus, we have the following theorem:

###### Theorem 3.3.

Let be a hyperbolic manifold that admits a totally geodesic immersion to a compact, right-angled Coxeter orbifold, , of dimension 3. Then for any , there exists a subgroup of such that , and index of is bounded above by

 2πVPsinh2(ln(√3+√2)+dP)ℓρ(α),

where is the length of the unique geodesic representative of , and where and are the diameter and volume of , respectively.

###### Proof.

We know that Vol so if consists of polyhedra,

 k<πVPsinh2(ln(√3+√2)+dP)ℓρ(α).

Let be one lift of to . By a lift of , we mean a lift of the path to a path starting at the basepoint . Using the right-angled tiling of , we can lift to so that the result is a connected, convex union of polyhedra in denoted by , which contains the geodesic segment . The convexity of the set is crucial since we will want to apply the Poincaré Polyhedron Theorem to prove the result above.

Set to be one of the two lifts of that share endpoints with . If is the associated convex lift of containing , then is a convex union of polyhedra in , such that one endpoint of is contained in the interior of .

Denote by the group of isometries of generated by reflections in the sides of . Then , and is a fundamental domain for the action of on by the Poincaré Polyhedron Theorem. Since contains polyhedra, . Letting be the covering map, we then have that the restriction of to the interior of is a homeomorphism onto its image in . Thus, is not a loop in , and .

Now, if , then and . The result follows. Note that we have used both the fact that and that the length of is equal to to its translation length in in a crucial way.

## 4. The Proof of Theorem 4.3

In this section we obtain the analogous results of the previous section for hyperbolic manifolds that admit a totally geodesic immersion to a compact, right-angled Coxeter orbifold of dimension 4. We again denote the diameter and volume of by and , respectively. As before, the images of under the action of tessellate . We have the following analog of Lemma 3.2:

###### Lemma 4.1.

Let be the analog in of from Lemma 3.2. That is to say is a tubular –neighborhood of the geodesic segment between the points and lying on the –axis in . Then Vol, where is the length of the geodesic between the points and in .

###### Proof.

To calculate the volume of we again find it convenient to use generalized spherical coordinates. In spherical coordinates are defined by:

 x1 =rsinϕ1sinϕ2sinϕ3 x2 =rsinϕ1sinϕ2cosϕ3 x3 =rsinϕ1cosϕ2 x4 =rcosϕ1,

where . Thus, the volume form on is

 dx1∧dx2∧dx3∧dx4x44=(1rtan2ϕ1sec2ϕ1sinϕ2)dr∧dϕ1∧dϕ2∧dϕ3.

As before, the range of values for in is , where is the angle in the plane from the positive -axis to the Euclidean ray consisting of points at hyperbolic distance from the -axis. Additionally, the range of values of and in are unrestricted so that takes values in and takes values in . Thus,

,

with the last equality coming from the angle of parallelism laws in hyperbolic space (see [Beardon, Section 7.9]).

Next we prove the analog of Lemma 3.1. As in the previous section, is the cyclic subgroup generated by , is , and is the unique simple closed geodesic in . With as the geodesic axis for , the preimage of under the covering map , we again denote the convexification of in by .

###### Lemma 4.2.

Let be the union of polyhedra forming the convexification of   in via the procedure mentioned above. Then any polyhedron must intersect , the –neighborhood of , where .

###### Proof.

As in the proof of Lemma 3.1 we let be the preimage of in , which forms an –neighborhood around . Suppose is a polyhedron in our tessellation of which does not intersect . We aim to show that a hyperplane in containing one of the codimension–1 faces of must separate from , demonstrating that and proving the lemma.

Let be the minimal-dimensional face of containing its closest point to and set codim. Again, is also the number of codimension–1 faces of that intersect . Take a shortest geodesic from to that intersects at a point and at a point . Then is an orthogeodesic with since does not intersect the –neighborhood of . We let be a hyperplane through that is perpendicular to , which separates from . Again necessarily contains .

We note that the proof in the cases are exactly the proofs for , respectively, of Lemma 3.1. We therefore consider the case where is a vertex of and . We begin by sending to the origin of via isometries. Recall that is the hyperplane of obtained by restricting the coordinate to zero for . Since is an all right polyhedron, the four codimension–1 faces of that intersect necessarily lie in four hyperplanes , , , and that are the isometric image of , , , and . Letting , we see that , , , and form an all right-angled spherical tetrahedron in . We are, therefore, interested in the threshold, , of the distance between and so that can be inscribed in an all right spherical tetrahedron formed by the boundaries of four pairwise orthogonal hyperplanes. Then if , at least one of the four two-spheres cannot intersect , and thus, one of the four hyperplanes must separate from .

We begin by calculating the radius of a sphere inscribed in such an all right tetrahedron as shown in Figure 7. As indicated by the figure, we calculate the radius using a spherical triangle formed by a midpoint of an edge of the tetrahedron, , a point of tangency of the inscribed sphere,, and the center of the sphere.

Note that the angle at is , the angle at is , and the length of the edge between and is the radius of a circle inscribed in a spherical triangle whose edge lengths are , which we calculated above to be . An application of a spherical law of cosines gives that the angle at is , and a second application gives that .

We now use the same cross sectional picture as in the proof of Lemma 3.1, which is shown is Figure 3. The angle is again equal to , where now . Therefore, and . It follows that and , which by a simple calculation gives us .

Taking the maximum over the four cases gives .

The proof of Theorem 4.3 now follows exactly as the proof of Theorem 3.3. Lemma 4.2 tells us that , where can be taken to be . We take a subset of the preimage of in that is isometric to and which forms a region like from Lemma 4.1, where . The lemma then implies that

 Vol (C)< Vol (NR+dP(¯¯¯¯α))=43πsinh3(R+dP)ℓρ(α)=43πsinh3(ln(2+√3)+dP)ℓρ(α).
###### Theorem 4.3.

Let be a hyperbolic manifold that admits a totally geodesic immersion to a compact, right-angled Coxeter orbifold, , of dimension 4.Then for any , there exists a subgroup of such that , and the index of is bounded above by

 8π3VPsinh3(ln(2+√3)+dP)ℓρ(α),

where is the length of the unique geodesic representative of , and where and are the diameter and volume of , respectively.

###### Proof.

We know that Vol so if consists of polyhedra,

 k<4π3VPsinh3(ln(2+√3)+dP)ℓρ(α).

We form the convex set as in the proof of Theorem 3.3. Thus, is the convex union of copies of containing one endpoint of a lift of to in its interior.

Let be the group of isometries of generated by reflections in the sides of . Then , and is a fundamental domain for the action of on by the Poincaré Polyhedron Theorem. Thus, the image of is not a loop in , and . Now, let . Then, and . The result follows.

An immediate corollary of Theorems 3.3 and 4.3 is:

###### Corollary 4.4.

Let be a hyperbolic manifold admitting a totally geodesic immersion to a compact, right-angled Coxeter orbifold of dimension 3 or 4. Then, the geodesic residual finiteness growth is at most linear. That is to say, .

## 5. Extension to Manifolds that Virtually Immerse into Compact Reflection Orbifolds

In this section we extend Corollary 4.4 to all hyperbolic manifolds that virtually admit a totally geodesic immersion to a compact, right-angled Coxeter orbifold of dimension 3 or 4. The key fact is the following lemma, which is the analog of [BHP, Lemma 2.2] for geodesic residual finiteness growth functions (rather than the usual residual finiteness growth functions, calculated with respect to word length).

###### Lemma 5.1.

Let be a hyperbolic –manifold and let be a finite index subgroup with . Let be the cover of of index corresponding to the subgroup . Then the geodesic residual finiteness function for is bounded by that of . That is to say, and hence .

###### Proof.

For an element , we see that

 {H≤π1(M):α∉H}⊇{K′≤K≤π1(M):α∉K′}.

Therefore,

 Dπ1(M)(α) =min{[π1(M):H]:α∉H,H≤π1(M)} ≤min{[π1(M):K′]:α∉K′,K′≤K≤π1(M)} =Cmin{[K:K′]:α∉K′,K′≤K}=C⋅DK(α),

where we set if . The equality above comes from the fact that .

Next, we claim that

 FM,ρ(n)= max{Dπ1(M)(α):α∈π1(M)−{1},ℓρ(α)≤n} ≤ max{C⋅DK(β):β∈K−{1},ℓρ