New York Journal of Mathematics

New York J. Math. 8 (2002) 169–179.

Equivalence of Geometric and Combinatorial Dehn Functions

José Burillo and Jennifer Taback

Abstract. We prove that if a finitely presented group acts properly discontinuously, cocompactly and by isometries on a simply connected Riemannian manifold, then the Dehn function of the group and the corresponding filling function of the manifold are equivalent, in a sense described below. We also prove this result for simplicial complexes X where the metric on X restricts to a Riemannian metric with corners on each simplex.

Contents

1. Dehn functions and their equivalence 169 2. Technical lemmas 171 3. Proof of the main theorem 175 References 178

1. Dehn functions and their equivalence

Let X be a simply connected 2-complex , and let w be an edge circuit in X^{(1)}. If D is a van Kampen diagram for w (see [5]), then the area of D is defined as the number of 2-cells on D, and the area of w, denoted a(w), is defined as the minimum of the areas of all van Kampen diagrams for w. The Dehn function of X is then defined to be

δ_{X}(n) = max a(w),

where the maximum is taken over all loops w of length l(w) ≤ n. Given two functions f and g from ℕ to ℕ (or, more generally, from ℝ^{+} to ℝ^{+}), we say that f ≺ g if there exist positive constants A, B, C, D, E so that

f(n) ≤ Ag(Bn + C) + Dn + E.

Two such functions are called equivalent (denoted f ≡ g) if f ≺ g and g ≺ f. The Dehn function is invariant under quasi-isometries; when one considers the 1-skeleton of a complex as a metric space with the path metric, where every edge has length one, two complexes with quasi-isometric 1-skeleta have equivalent Dehn functions

169^{ISSN 1076-9803/02}

Received January 15, 2001.

Mathematics Subject Classification. Primary 20F65; secondary: 20F05, 20F06, 49Q15.

Key words and phrases. Dehn function, van Kampen diagram.

(see [1]). Let G be a finitely presented group, and let P be a finite presentation for G. Let K = K(P) be the 2-complex associated to P, i.e., the 2-complex with a single vertex, an oriented edge for every generator of P, and a 2-cell for every relator, attached to the edges according to the spelling of the relator. Then the Dehn function of P is, by definition, the Dehn function δK̃ of the universal covering of K. Two finite presentations P and Q for the same group G yield 2-complexes K̃(P) and K̃(Q) with quasi-isometric 1-skeleta, and hence equivalent Dehn functions. Thus the Dehn function of the group G is defined to be the equivalence class of the Dehn function of any of its presentations. An extensive treatment of Dehn functions of finitely presented groups is given in [4]. A closely related definition can be formulated in the context of Riemannian manifolds, dating back to the isoperimetric problem for ℝ^{n} in the calculus of variations. Given a Lipschitz loop γ in a simply connected Riemannian manifold M, we define the area of γ to be the infimum of the areas of all Lipschitz discs bounded by γ. We then define the geometric Dehn function of M by

δ_{M}(x) = max_{l(γ)≤x} area(γ)

where l(γ) represents the length of γ. It is natural to consider the question of whether the Dehn functions of a simply connected Riemannian manifold M and of a finitely presented group G acting properly discontinuously and cocompactly by isometries on M agree. The fact that they effectively agree has been implicitly assumed in the literature, though no proof has been given. A closely related statement is given in [2, Theorem 10.3.3], applying the Deformation Theorem of Geometric Measure Theory ([3, 4.2.9] and [8]) to this setting, and which provides the basis of the Pushing Lemma below. This paper is devoted to providing a complete and detailed proof that the combinatorial and geometric Dehn functions are equivalent. It is known to the authors that M. Bridson has lectured on an alternate, unpublished proof of the same result. The authors would like to thank Professor S. M. Gersten for his encouragement and his useful remarks, Kevin Whyte for helpful conversations and the referee for his precise comments.

We state our result below not only for Riemannian manifolds, but for simplicial complexes with certain metric properties, to obtain the greatest generality. We require that the metric on the simplicial complex restricts on each simplex to give the structure of a Riemannian manifold with corners. Recall that each point p in a manifold with corners has a neighborhood diffeomorphic to a neighborhood of the origin in ℝ_{+}^{k} × ℝ^{n−k}, for appropriate integers n and k, where the point p maps to the origin. For a complete definition of a manifold with corners, we refer the reader to [6]. Clearly, for such a simplicial complex, the Dehn function is well-defined, since a loop in it will intersect only finitely many simplices, and the length of a loop is computed by summing the lengths of its component parts in each simplex. Similarly, the area of a disc is well-defined, so we can consider the Dehn function of such a complex. Once a triangulation is defined on a Riemannian manifold, it trivially becomes a simplicial complex of this type, where the Riemannian structure on each simplex is induced by the global structure of the manifold. The main advantage of using these complexes is that our result applies to spaces which are not topologically manifolds.

Equivalence of Geometric and Combinatorial Dehn Functions 171

Theorem 1.1. Let M be a simply connected simplicial complex whose metric restricts on each simplex to give the structure of a Riemannian manifold with corners. Let G be a finitely presented group acting properly discontinuosly, cocompactly and by isometries on M. Let τ be a triangulation of M, i.e., a G-invariant subdivision of the simplicial structure for M. Then the following three Dehn functions are equivalent:

(1) the Dehn function δ_{G} of any finite presentation of G,

(2) the Dehn function δ_{τ^{(2)}} of the 2-skeleton of τ, and

(3) the Dehn function δ_{M} of M.

When a Riemannian manifold is triangulated, then it acquires the structure of a simplicial complex with the stipulated metric condition. Hence, as an immediate corollary of this theorem, we obtain the following result.

Corollary 1.2. Let M be a simply connected Riemannian manifold and let G be a finitely presented group acting properly discontinuosly, cocompactly and by isometries on M. Then the geometric Dehn function of M and the combinatorial Dehn function of G are equivalent.

The fact that δ_{G} and δ_{τ^{(2)}} are equivalent is clear: since G acts cocompactly on τ, there is a quasi-isometry between τ^{(1)} and the 1-skeleton of K̃(P) for any presentation P of G, and the equivalence follows from the results in [1]. We will concentrate on the proof of the equivalence between δ_{τ^{(2)}} and δ_{M}. The arguments will be mainly geometric, relating the lengths and areas of loops and discs in M with those included in the triangulation τ. The first step in this direction is the Pushing Lemma, which is a complete analog of the Deformation Theorem in Geometric Measure Theory and already stated and proved, in a slightly different way, in [2, Theorem 10.3.3], and whose proof we will follow closely.

2. Technical lemmas

The Pushing Lemma, stated below, will allow us to relate arbitrary Lipschitz chains in M to chains in the corresponding skeleta of τ. The main technical problem to be overcome is that projection of a Lipschitz chain to τ from a badly chosen point can increase the volume of the chain arbitrarily. We overcome this by using techniques from measure theory that assure the existence of a center of projection far enough from the chain, thus providing control on the growth of the volume.

Lemma 2.1 (Pushing Lemma). Let M, G and τ be as above. Then there exists a constant C, depending only on M and τ, with the following property: Let T be a Lipschitz k-chain in M, such that ∂T is included in τ^{(k−1)}. Then there exists another Lipschitz k-chain R, with ∂R = ∂T, which is included in τ^{(k)}, and a Lipschitz (k + 1)-chain S, with ∂S = T − R, satisfying

vol_{k}(R) ≤ Cvol_{k}(T)  and  vol_{k+1}(S) ≤ Cvol_{k}(T).

In particular, if T is a loop, so is R, and S is a homotopy from T to R.

The Pushing Lemma differs from the statement in [2] because it applies to chains as well as cycles, since the boundary of the chain is not modified, as it is included in the (k − 1)-skeleton. A statement for cycles is not sufficient, since this lemma will be applied to chains as well as loops, and the fact that ∂T = ∂R is crucial in the proof of the main theorem. We first prove a lemma which will later allow us to choose our center of projection to lie away from the Lipschitz chain T.

Lemma 2.2. Let f : S^{k} → σ_{k+1} be Lipschitz with constant L, where σ_{k+1} is the standard Euclidean (k +1)-simplex. Then f(S^{k}) has Lebesgue (k +1)-measure zero.

Proof. Since S^{k} is compact, choose a finite open cover of S^{k} by k-dimensional balls B_{i} of radius \frac{1}{n}. We can cover S^{k} with C_{1}n^{k} such balls, for some constant C_{1}. The image of any ball B_{i} under the Lipschitz map f is contained in a (k+1)-dimensional ball B′_{i} ⊂ σ_{k+1} with (k + 1)-volume \frac{C_{2}}{n^{k+1}} for some constant C_{2}. Then the total volume of the collection \{B′_{i}\} is at most \frac{C_{1}C_{2}}{n}. So f(S^{k}) is contained in an open set of σ_{k+1} whose total volume is \frac{C_{1}C_{2}}{n} and thus f(S^{k}) has Lebesgue measure 0. □

Proof of Lemma 2.1. The proof will proceed by descending induction on the skeleta of τ. Assume that a Lipschitz k-chain T is included in τ^{(i)} but not in τ^{(i−1)}, for i > k. We want to proceed simplex by simplex, choosing an appropriate point not in T in each simplex and projecting the chain T radially from this point to the boundary of the simplex. We will prove the following claim.

Claim: There exists a constant C with the property that for every simplex, there is a point p not in T so that radial projection of T from p to the boundary of the simplex does not increase the volume of the chain by more than a multiplicative factor of C.

Observe that since T is compact, it only intersects finitely many simplices of τ, and in each simplex is only modified by a radial projection from a point not in T. These radial projections increase the Lipschitz constant of T, but the chain R obtained after the projections will still be Lipschitz. To simplify the computations, we will work through the proof in the unit Euclidean simplex. Since G acts cocompactly on M, we can construct a sufficiently fine finite triangulation of the quotient and lift it to M. If the simplices are small enough we can map them to ℝ^{n} via the exponential map. Since the exponential map is Lipschitz, the changes in the metric are bounded by only a multiplicative constant. We then have a finite number of simplices in ℝ^{n}, so the distortion is again bounded. Thus working with the unit simplex only affects the value of the constant C.

Let σ be the unit Euclidean i-simplex, O the barycenter of σ, and r a positive number so that the ball of center O and radius 3r is included in the interior of σ. Let B be the ball of center O and radius r, with u an element of B, and B_{u} the ball of center u and radius 2r. Clearly B ⊂ B_{u}, for all u. Let π_{u} be the radial projection with center u of B_{u} \ {u} onto ∂B_{u}. Let Q = T ∩ σ. We want to see that there exists a constant v_{0} independent of T and σ, and a point u ∈ B \ Q, dependent on T, with

vol_{k}(π_{u}Q) ≤ v_{0} vol_{k}(Q).

From Lemma 2.2, we see that the set B \ Q has the same measure as B, allowing us to choose u ∈ B \ Q. For every positive real number v define

A_{v} = {u ∈ B \ Q | vol_{k}(π_{u}Q) > v vol_{k}(Q)}

and let α(v) = m_{i}(A_{v}), where m_{i} is the i-dimensional Lebesgue measure. We want to prove that

lim_{v→∞} α(v) = 0.

Then we will choose v_{0} with α(v_{0}) < m_{i}(B), so the measure of A_{v_{0}} will be less than the measure of B. Thus there will exist a point u ∈ (B \ Q) \ A_{v_{0}}, which will be the center of projection. Since u ∉ A_{v_{0}}, this projection will increase the area at most by a multiplicative factor v_{0}.

............................................................................................................................................................................................................................................ ...........................................................................................................................................................................................................................................................................................................................  ........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ ........................................................................ ................................................  ...................................... ..............................................................................................................................................................  σ O uB B_{u} Q π_{u}Q  . . Figure 1: Projecting Q to the boundary of B_{u}.

We have

vol_{k}(π_{u}Q) ≤ vol_{k}(π_{u}(Q ∩ B_{u})) + vol_{k}(Q)

≤ ∫_{Q∩B_{u}} (\frac{2r}{||x − u||})^{k} dx + vol_{k}(Q),

where the first term accounts for the volume obtained after projecting, and the second term takes care of the possibility of Q and B_{u} being disjoint. Assume now that vol_{k}(Q) is nonzero (if vol_{k}(Q) = 0 then vol_{k}(π_{u}Q) = 0). Then we have:

α(v) v vol_{k}(Q) = v vol_{k}(Q) ∫_{A_{v}} du = ∫_{A_{v}} v vol_{k}(Q) du

≤ ∫_{A_{v}} vol_{k}(π_{u}Q) du ≤ ∫_{B} vol_{k}(π_{u}Q) du

≤ ∫_{B} (∫_{Q∩B_{u}} (\frac{2r}{||x − u||})^{k} dx + vol_{k}(Q)) du

= (2r)^{k} ∫_{Q∩B_{u}} ∫_{B} ||u − x||^{−k} du dx + vol_{i}(B)vol_{k}(Q).

Notice that the function (\frac{2r}{||x−u||})^{k} is bounded above and below, since u ∉ Q ∩ B_{u}, and is integrated over compact regions. This allows us to change the order of integration. Now make a change of variables, letting w = u − x, and increase the domain of integration to B(0, 3r). We continue with the upper bound for α(v) v vol_{k}(Q):

α(v) v vol_{k}(Q) ≤ (2r)^{k} ∫_{Q∩B_{u}} ∫_{B} ||u − x||^{−k} du dx + vol_{i}(B)vol_{k}(Q)
≤ (2r)^{k} ∫_{Q∩B_{u}} dx ∫_{B(O,3r)} ||w||^{−k} dw + vol_{i}(B)vol_{k}(Q)
≤ Kvol_{k}(Q),

where

K = (2r)^{k} ∫_{B(O,3r)} ||w||^{−k} dw + vol_{i}(B).

Observe that K is finite and independent of T and σ. We conclude that α(v)v ≤ K. Knowing K, we can find v_{0} such that K/v_{0} < m_{i}(B), where v_{0} is a constant independent of T and σ. We have now found A_{v_{0}} with strictly less measure than B, and can pick a point in (B \ Q) \ A_{v_{0}} from which to project so that the volume increases at most by a multiplicative factor v_{0}. The result of the above argument is the construction of another chain π_{u}Q which is far enough from O. We can now project radially from O to ∂σ, and the change of volume is bounded since π_{u}Q is at least at a distance r from O. The combination of this change of volume with v_{0} gives the constant needed in this precise skeleton. Combining the constants from all of these steps, we obtain the desired constant C. Observe that these projections leave τ^{(i−1)} unchanged, so clearly ∂T is preserved. The (k + 1)-chain S is obtained by joining every x ∈ Q to π_{u}x by a segment. The volume of the piece of S contained in σ is then bounded, as before, by

(2r)^{k+1} ∫_{Q∩B_{u}} \frac{dx}{||x − u||^{k}} ,

where the extra factor 2r is obtained from the direction of the projection, since each segment has length bounded by 2r. An argument similar to the previous one shows that projecting from most points in B gives the correct bound for the volume. □

The third lemma states that for a Lipschitz map, almost every point in the target space has a finite number of preimages. It is a direct consequence of the area formula for Lipschitz maps, and it will be used in the proof of Theorem 1.1.

Lemma 2.3. Let A, B ⊂ ℝ^{k} be open sets, and assume that vol_{k}A is finite. Let f : A → B be a Lipschitz map. Then the set of points in B with infinite preimages under f has Hausdorff k-measure zero.

Proof. Since f is a Lipschitz map, by Rademacher’s Theorem ([3, 3.1.6]) it is differentiable almost everywhere (with respect to the Lebesgue k-measure), so the Jacobian J_{k}f(x) is well-defined for almost all x ∈ A. Observe that our notation is slightly different from the one in [3], because J_{k}f(x) denotes the Jacobian here, but the absolute value of the Jacobian in [3]. For y ∈ B, let N(f, y) be the number of elements of f^{−1}(y), possibly infinite, and denote by m_{k} and h_{k} the Lebesgue and Hausdorff k-measures, respectively. Then the area formula for Lipschitz maps ([3, 3.2.3]) states that

∫_{A} |J_{k}f(x)| dm_{k}(x) = ∫_{M} N(f, y) dh_{k}(y).

Since f is Lipschitz, we know that |J_{k}f(x)| is bounded, and since A has finite volume, the integral on the left-hand side is finite. So the set where N(f, y) is infinite cannot have positive Hausdorff k-measure, because then the right-hand side of the equation would be infinite. □

3. Proof of the main theorem

We begin by proving the one of the two inequalities necessary for the equivalence of δ_{M} and δ_{τ^{(2)}}, namely

δ_{M} ≺ δ_{τ^{(2)}}. (3.1)

Let γ be a Lipschitz loop in M, with length at most n. Using the Pushing Lemma, we can construct a new loop η, of length at most Cn, which is included in the 1skeleton, and the homotopy between γ and η has area at most Cn. The loop η is not necessarily combinatorial, but it is a rectifiable loop in a nonpositively curved space, namely the metric graph τ^{(1)}. So there is a unique (up to reparametrization) closed geodesic ζ in the free homotopy class of η. The straight homotopy (in τ^{(1)}) from η to ζ is a map from an annulus to τ^{(1)}. The length of ζ decreases monotonically and its area can be made arbitrarily small. The combinatorial loop ζ can be filled combinatorially by at most δ_{τ^{(2)}}(Cn) 2-simplices in τ. Thus

δ_{M}(n) ≤ Aδ_{τ^{(2)}}(Cn) + 2Cn,

where A is the area of the largest 2-cell in τ, and it follows that δ_{M} ≺ δ_{τ^{(2)}}. To prove the reverse inequality

δ_{τ^{(2)}} ≺ δ_{M},

to (3.1), we start with a combinatorial loop γ in the 1-skeleton of τ, with length at most n. Let

f : D^{2} ⟶ M

be a Lipschitz disc in M with boundary γ, and with area a. We want to construct a van Kampen diagram for γ and bound its area in terms of a. The first step is to use the Pushing Lemma to find a new disc (also denoted f) which is included in τ^{(2)}, and whose area is at most Ca. So assume that f(D^{2}) is included in the 2-skeleton of τ. We can also assume Ca is not zero, because if the area of f is zero, there is no problem making it combinatorial and its area remains zero, satisfying the inequalities trivially. So there exists an open 2-simplex σ of τ such that σ ∩ f(D^{2}) has strictly positive area. Observing that both σ and f^{−1}(σ) are open sets, and that f^{−1}(σ) has finite area, since it lies in D^{2}, so we can apply Lemma 2.3 to the map

f|f^{−1}(σ) : f^{−1}(σ) ⟶ σ.

We conclude that the set of points with infinite preimages under f|f^{−1}(σ) has

measure zero, and, since the area of the image is strictly positive by the choice of σ, we can choose a point p ∈ σ such that f^{−1}(p) is finite.

Let X be a component of f^{−1}(σ). Note that X is, clearly, an open set of D^{2}, so it is a manifold itself. If X ∩ f^{−1}(p) = ∅, then f|X can be modified by composing with a radial projection from p. After this change, a component X of f^{−1}(σ) satisfies X ∩ f^{−1}(p) ≠ ∅, and there are only finitely many of these components. Moreover, if f|X is not surjective, we can again modify f|X by a radial projection from a point not in f(X), to push its image to ∂σ. After these changes to f, there is a component X of f^{−1}(σ) so that f|X is surjective, and X ∩ f^{−1}(p) ≠ ∅. If X is one such component, the original f has not been modified in X by any radial projection, and the map

f|X : X ⟶ σ

is still Lipschitz, since it is the restriction of the original map f. We will obtain a lower bound on the area of f|X using the degree of f|X. Observe that f|X is a map between two manifolds, so we can apply Rademacher’s theorem to it and conclude that it is differentiable almost everywhere. Consequently, we can define the degree of f|X at a point y ∈ f(X) by

deg f|X(y) = ∑_{x∈f^{−1}(y)} sign J_{2}f(x).

Moreover, since X is an open connected component of f^{−1}(σ), we have that f(X) ⊂ σ and f(∂X) ⊂ ∂σ, so f(X) and f(∂X) are disjoint. Then, by [3, 4.1.26], the degree of f|X is almost constant in f(X), and we can define the degree of f|X as the value d_{X} it achieves at almost every y ∈ f(X). The lower bound on the area of f|X is given by the area formula for Lipschitz maps: if u is an integrable function with respect to m_{2}, we have (see [3, 3.2.3]):

∫_{X} u(x)|J_{2}f(x)| dm_{2} = ∫_{σ} ∑_{x∈f^{−1}(y)∩X} u(x) dh_{2},

and taking u(x) = sign Jf(x) we obtain:

area f|X = ∫_{X} |J_{2}f(x)| dm_{2} ≥ |∫_{X} J_{2}f(x) dm_{2}|

= |∫_{X} sign J_{2}f(x) |J_{2}f(x)| dm_{2}|

= |∫_{σ} deg f|X dh_{2}| = \frac{\sqrt{3}}{4}|d_{X}|.

Our goal is to find a simplicial map

g : D^{2} ⟶ τ^{(2)}

(with some simplicial structure in D^{2}) such that only |d_{X}| simplices are mapped by the identity to σ under g|X, and the rest of X is mapped to ∂σ. Then we will have that the combinatorial area of g is bounded as follows:

∑_{X} |d_{X}| ≤ ∑_{X} \frac{4}{\sqrt{3}}area (f|X) ≤ \frac{4}{\sqrt{3}}Ca,

giving us the required bound. Note that the map g is not combinatorial, but only simplicial, and at the end of the proof a short argument will be required to ensure the existence of a combinatorial map whose area admits the same upper bound. The first step in finding the map g is to smooth the map f|X, in order to apply differentiable techniques to it. Let O be the barycenter of σ, and choose 0 < ϵ < r such that:

∅ ≠ B(O, r − ϵ) ⊂ B(O, r) ⊂ B(O, 2r) ⊂ B(O, 2r + ϵ) ⊂ σ,

and let U_{1} = f^{−1}(B(O, r)) and U_{2} = f^{−1}(B(O, 2r)). We have that \overline{U_{1}} ⊂ U_{2} ⊂ \overline{U_{2}} ⊂ X. Choose δ > 0 so that B(x, δ) ⊂ X for all x ∈ U_{2}, and so that if |x − y| < δ then |f(x) − f(y)| < ϵ, for all x, y ∈ X. Let φ be a C^{∞} bump function in ℝ^{2} with support in B(0, δ), and with integral 1. Then, for x ∈ U_{2}, we can construct the convolution

f ∗ φ(x) = ∫_{B(x,δ)} f(x − z)φ(z) dz,

which is C^{∞} in U_{2}, and satisfies |f(x) − f ∗ φ(x)| < ϵ for all x ∈ U_{2}. Also, if f|X was Lipschitz with constant L, then f ∗ φ is also Lipschitz with the same constant: if x, y ∈ U_{2},

|f ∗ φ(x) − f ∗ φ(y)| ≤ |f(x − z) − f(y − z)| ∫_{B(0,δ)} φ(z) dz ≤ L|x − y|.

Now choose a Lipschitz function α on X with values in [0, 1] which is equal to 1 in U_{1} and equal to 0 outside U_{2}, and define

f̃ = α(f ∗ φ) + (1 − α)f|X.

Note that f̃ is defined only on X. Then f̃ satisfies the following properties: (1) f̃ is defined as a map from X into σ, which are both manifolds,

(2) |f(x) − f̃(x)| < ϵ for all x ∈ X,

(3) f̃ is smooth in U_{1},

(4) f̃ = f in X \ U_{2},

(5) f̃ is Lipschitz, and

(6) deg f̃ = deg f|X. The first four properties are clear from the construction of f̃, and property (5) holds because f|X and f ∗φ and α are all Lipschitz. To see that the degree is unchanged, since the degree of f̃ is almost constant (recall that X and σ are manifolds), and f|X and f̃agree outside U_{2}, we only need to find a point in σ\B(O, 2r+ϵ) for which the degree is d_{X} for both f|X and f̃. Again, using the fact that f̃is a map between manifolds, we can use Sard’s Theorem ([7]) to claim the existence of a regular value for f̃ in B(O, r − ϵ) whose preimages are all in U_{1}. Let q be this regular value and let p_{1}, …, p_{m} be its preimages. Let V be an open disc with center q such that f̃^{−1}(V) = V_{1} ∪ ⋯ ∪ V_{m}, where the V_{i} are discs around p_{i}, pairwise disjoint, and such that f̃|V_{i} is a diffeomorphism. In general, we will have that m > |d_{X}|, and must cancel discs with opposite orientations. Assume V_{m−1} and V_{m} are mapped to V with opposite orientations. Choose a ∈ ∂V_{m−1} and a′ ∈ ∂V_{m} with f̃(a) = f̃(a′), and join a and a′ with a simple path λ such that f̃(λ) is nullhomotopic in σ \ V. This can be done because the map

f̃: X \ ⋃_{i=1}^{m} V_{i} ⟶ σ \ V

induces a surjective homomorphism of fundamental groups. After contracting f̃(λ), we can assume f̃(λ) is the constant path f̃(a). Remove the discs V_{m−1} and V_{m} and perform surgery along λ. The new boundary thus created is mapped to ∂V under f̃ by a map from S^{1} to itself of degree zero. Extend this map to a map from D^{2} to S^{1} and attach it to f̃ along this boundary. For the new map (which we will continue calling f̃), the preimage of q consists only of the points p_{1}, …, p_{m−2}. Repeating this process we will obtain a map where now only the discs V_{1}, …, V_{|d_{X}|} are mapped to V, all with the same orientation. Choose (temporarily) a sufficiently fine subdivision of τ so that there is a 2-simplex W in V, and let ρ_{i} = f̃^{−1}(W). Modify the map in X by composing with the expansion of W into all of σ.

........................................ ............................................... ...................................................... ............................................................ ............................................................ ...................................................... .................................. ........................... .................... ........................... ................................. ........................................ ............................................... ..................................................... ..................................................... ..................................................... ............................................... .................... .............. ........................... ................................. ........................................ ............................................... ............................................... ........................................ ................................. ................................. ................................. ........................... .................... ...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................  ........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................  ....................................................................................................................................................................................................................................................................................................................................................................................................... ............................................................................. ...................  ............................................................................. ................... .......................................................................................................................................................  ....................................................................................................................................................... .......................................................................................................................................................  X ρ_{1}  ρ_{2} ρ_{3}  Wσ  σ Figure 2: Making the map f simplicial

After this process is done for all σ, we obtain a map from D^{2} to τ^{(2)}, where all the ρ_{i} are sent homeomorphically to 2-simplices of τ, and the rest of D^{2} is sent to the 1-skeleton of τ. To finish the construction of g, find a simplicial structure on D^{2} compatible with the simplicial structure on the original loop γ and which includes all the ρ_{i} obtained for all σ as 2-simplices. Now approximate the map f̃ simplicially within τ^{(1)} relative to all the ρ_{i} and to γ. The result is simplicial, and the number of simplices sent by g homeomorphically to 2-simplices in τ is

∑_{X} |d_{X}| ≤ \frac{4}{\sqrt{3}}Ca.

This map is not a van Kampen diagram yet, since it is only simplicial. To finalize the proof of the inequality

δ_{τ^{(2)}} ≺ δ_{M},

we will find a van Kampen diagram which satisfies the same upper bound as the map g. Consider simplicial maps from a contractible planar 2-complex Y into τ^{(2)}, with boundary γ, whose area satisfies the same bound as g. (The map g shows the existence of such maps.) Among all these maps, choose one with the minimum number of 2-cells in Y. This map is necessarily combinatorial, since if some 2-cell of Y is collapsed to the 1-skeleton of τ^{(2)}, we could collapse it in Y and find a map with fewer 2-cells. This map is the required van Kampen diagram for the loop γ, and the second inequality is proved.

References

[1] J. M. Alonso, Inégalités isopérimétriques et quasi-isométries, C. R. Acad. Sci. Paris, Série I, 311 (1990), 761–764, MR 91k:57004, Zbl 0726.57002.

[2] D. B. A. Epstein, J. W. Cannon, D. F. Holt, S. V. F. Levy, M. S. Paterson and W. P. Thurston, Word Processing in Groups, Jones and Bartlett, Boston-London, 1992, MR 93i:20036, Zbl 0764.20017.

[3] H. Federer, Geometric Measure Theory, Springer-Verlag, New York-Berlin-Heidelberg, 1969, MR 41 #1976, Zbl 0874.49001.

[4] S. M. Gersten, Dehn functions and l_{1}-norms of finite presentations, Algorithms and Classification in Combinatorial Group Theory (G. Baumslag and C.F. Miller III, eds.), Math. Sci. Res. Inst. Publ., 23, Springer-Verlag, New York-Berlin-Heidelberg, 1992, 195–224, MR 94g:20049, Zbl 0805.20026.

[5] R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 89, Springer-Verlag, New York-Berlin-Heidelberg, 1977, MR 58 #28182, Zbl 0368.20023.

[6] Richard B. Melrose, Geometric Analysis on Manifolds with Corners, book draft.

[7] J. W. Milnor, Topology from the Differentiable Viewpoint, University Press of Virginia, Charlottesville, 1965, MR 37 #2239, Zbl 0136.20402.

[8] L. Simon, Lectures on Geometric Measure Theory, Proceedings of the Centre for Mathematical Analysis, Volume 3, Australian National University, 1983, MR 87a:49001, Zbl 0546.49019.

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain burillo@mat.uab.es Current address: Universitat Politecnica de Catalunya, Castelldefels (Barcelona), Spain burillo@mat.upc.es

Dept. of Mathematics and Statistics, University at Albany, Albany, NY 12222 jtaback@math.albany.edu http://math.albany.edu:8000/˜ jtaback

This paper is available via http://nyjm.albany.edu:8000/j/2002/8-11.html.