A note on $p$-adic simplicial volumes

We define and study generalizations of simplicial volume over arbitrary seminormed rings with a focus on $p$-adic simplicial volumes. We investigate the dependence on the prime and establish homology bounds in terms of $p$-adic simplicial volumes. As the main examples we compute the weightless and $p$-adic simplicial volumes of surfaces. This gives a way to calculate classical simplicial volume of surfaces without hyperbolic straightening and shows that surfaces satisfy mod $p$ and $p$-adic approximation of simplicial volume.


Introduction
The simplicial volume of an oriented compact connected manifold is the ℓ 1 -seminorm of the fundamental class in singular homology with R-coefficients, which encodes topological information related to the Riemannian volume [10]. A number of variations of simplicial volume such as the integral simplicial volume or weightless simplicial volume over finite fields proved to be useful in Betti number, rank gradient, and torsion homology estimates [8,20,14,15,16].
In the present article, we will focus on p-adic simplicial volumes. The basic setup is as follows: If M is an oriented compact connected manifold and (R, |·|) is a seminormed ring (see Section 2.1), then the simplicial volume of M with R-coefficients is defined as the infimum M, ∂M R := inf k j=1 |a j | k j=1 a j · σ j ∈ Z(M, ∂M ; R) ∈ R ≥0 over the "ℓ 1 -norms" of all relative fundamental cycles of M . For R or Z with the ordinary absolute value one obtains the classical simplicial volume M and the integral simplicial volum M Z . For a ring R with the trivial seminorm this gives rise to the weightless simplicial volume M (R) [16]. For other seminormed rings one obtains new, unexplored invariants. We prove a number of fundamental results that describe how these simplicial volumes for different seminormed rings are related.
Using the ring Z p of p-adic integers or the field Q p of p-adic numbers with the p-adic absolute value as underlying seminormed rings leads to p-adic simplicial volumes. The long-term hope is that M, ∂M Zp and M, ∂M Qp might contain refined information on p-torsion in the homology of M . We prove this in Section 3.3. While the inequalities M, ∂M (Fp) ≤ M, ∂M Zp and M, ∂M Qp ≤ M, ∂M Zp hold for all prime numbers p (see Corollary 2.10) and · (Fp) and · Zp exhibit similar behaviour, we are currently not aware of a single example where one of these inequalities is strict. Is a strict inequality related to p-torsion in the homology of M ?
1.3. Surfaces and approximation. In Section 4, we compute the p-adic simplicial volumes for some examples. In particular, we compute the weightless simplicial volume of surfaces. Let Σ g be the oriented closed connected surface of genus g. For b ≥ 1 we write Σ g,b to denote the surface of genus g with b boundary components.
Using this result we compute the Z p -simplicial volume of all surfaces (Corollary 4.6) and we give a new way to compute the classicial simplicial volume of surfaces, which avoids use of hyperbolic straightening (Remark 4.7). Moreover, Theorem 1.4 also shows that surfaces satisfy mod p and p-adic approximation of simplicial volume (Remark 4.8).

Non-values.
Recent results show that classical simplicial volumes are right computable [13], which in particular allows to give explicit examples of real numbers that cannot occur as the simplicial volume of a manifold. Based on the same methods we establish that also the p-adic simplicial volumes M Zp and M Qp are right computable; see Proposition 5.2.
Acknowledgements. C.L. was supported by the CRC 1085 Higher Invariants (Universität Regensburg, funded by the DFG).

Foundations
We introduce simplicial volumes with coefficients in rings with a submultiplicative seminorm, e.g., an absolute value. In particular, we obtain p-adic versions of simplicial volume. Moreover, we establish some basic inheritance and comparison properties of such simplicial volumes similar to those already known in the classical or weightless case.
2.1. Simplicial volume. Let R be a commutative ring with unit. A seminorm on R is a function | · | : R → R ≥0 with |1| = 1 that is submultiplicative |st| ≤ |s||t| and satisfies the triangle inequality |s + t| ≤ |s| + |t| for all s, t ∈ R. If the seminorm is multiplicative, it is called an absolute value. A seminormed ring is pair (R, |·|), consisting of a commutative ring R with unit and a seminorm | · | on R. A seminormed ring (R, | · |) is a normed ring if | · | is an absolute value.
Seminormed rings give rise to a notion of simplicial volume: Definition 2.1 (simplicial volume). Let (R, | · |) be a seminormed ring. Let M be an oriented compact connected d-manifold, and let Z(M, ∂M ; R) ⊂ C d (M ; R) be the set of all relative singular R-fundamental cycles of (M, ∂M ). Then the simplicial volume of M with R-coefficients is defined as Example 2.2 (classical simplicial volume). The usual norm on R is an absolute value on R. The corresponding simplicial volume · := · R is the classicial simplicial volume, introduced by Gromov [18,10]. Similarly, the usual norm on Z is an absolute value on Z. The corresponding simplicial volume is denoted by · Z , the so-called integral simplicial volume. Integral simplicial volume admits lower bounds in terms of Betti numbers [17,Example 14.28], logarithmic homology torsion [20], and the rank gradient of the fundamental group [15]. Example 2.3 (weightless simplicial volume). Every non-trivial commutative unital ring R can be equipped with the trivial seminorm The simplicial volume corresponding to the trivial seminorm will be called weightless and will be denoted by M, ∂M (R) . The weightless simplicial volume over finite fields R = F p has been studied before [16].
Example 2.4 (p-adic simplicial volumes). Let p be a prime number. The ring of p-adic integers is denoted by Z p and the field of p-adic numbers by Q p . The usual p-adic absolute value gives rise to two (possibly distinct) notions of p-adic simplicial volume, namely, · Zp and · Qp , respectively. Lemma 2.5. Let (R, | · |) be a seminormed ring and let I ⊂ R be a proper ideal. We define |s + I| R/I := inf{|s + i| | i ∈ I} for all s + I ∈ R/I. Then | · | R/I is a seminorm on the quotient ring R/I.

Proof.
We verify the submultiplicativity. Indeed, for all s, t ∈ R one obtains The triangle inequality follows from a similar argument.
Example 2.6 (seminorms on Z/p m Z). There are two distinct seminorms on the rings Z/p m Z that will play a role in this article. Using Lemma 2.5, the p-adic absolute value | · | p on Z p induces a seminorm on Z/p m Z. We will also denote this seminorm by | · | p ; for x = 0 it is given by x lies in p r Z/p m Z but not in p r+1 Z/p m Z. The corresponding simplicial volume will be denoted by · Z/p m Z .
As in Example 2.3 the rings Z/p m Z can be equipped with the trivial seminorm, which induces the weightless simplicial volume · (Z/p m Z) .

2.2.
Changing the seminorm. Let R be a commutative ring with unit. We denote by S(R) the set of all seminorms on R. We equip the space of all seminorms with the topology of pointwise convergence, for which a basis of open neighbourhoods of a seminorm α is given by the sets where ε ∈ R >0 and F is a finite subset of R.  Proof. Let α ∈ S(R) and let ε > 0. Take a relative fundamental cycle c = k j=1 a j σ j ∈ Z(M, ∂M ; R) with |c| α,1 < M, ∂M α + ε/2. Now every seminorm β ∈ U pw (ε/2k, {a 1 , . . . , a k }) satisfies M, ∂M β ≤ |c| β,1 ≤ |c| α,1 + ε/2 < M, ∂M α + ε and we deduce that the simplicial volume is upper semi-continuous with respect to the topology of pointwise convergence.

Changing the coefficients.
Proposition 2.8 (monotonicity). Let (R, | · | R ) and (S, | · | S ) be seminormed rings and let f : R −→ S be a unital ring homomorphism that for some λ > 0 satisfies |f (x)| S ≤ λ · |x| R for all x ∈ R. Then M, ∂M S ≤ λ · M, ∂M R holds for all oriented compact connected manifolds M .
Proof. As f is unital, the chain map C * (Id M ; f ) : C * (M ; R) −→ C * (M ; S) induced by f maps relative R-fundamental cycles to relative S-fundamental cycles of (M, ∂M ). Moreover,  Proof. It follows from the triangle inequality that the canonical unital ring homomorphism Z −→ R satisfies the hypotheses of Proposition 2.8 with the factor λ = 1. Proof. For the first three assertions we only need to apply Proposition 2.8 with λ = 1 (and Corollary 2.9) to the canonical projections and to the canonical inclusion Z p −→ Q p . The last assertion follows from Proposition 2.8 and the inequalities between the p-adic and the trivial seminorm on the ring Z/p m Z.
Taking ε → 0 proves the claim. Proof. By definition Z is |·| p -dense in Z p and Q is |·| p -dense in Q p . Therefore, we can apply Proposition 2.11.
In fact, we have the following simultaneous approximation result: Corollary 2.13. Let M be an oriented compact connected manifold and let T be a finite set of prime numbers.
Proof. For every prime p ∈ T we pick (using Corollary 2.12) a relative fundamental cycle c p ∈ Z(M, ∂M ; Z) that almost realizes the p-adic simplicial volume. The density of Z in p∈T Z p [19, (3.4)] allows us to find integers a p ∈ Z (for p ∈ T ) with p∈T a p = 1 and such that a p is close to 1 in the p-adic absolute value, but close to 0 in the q-adic absolute value for all q ∈ T \ {p}. Then c = p∈T a p c p is a relative fundamental cycle and approximates the p-adic simplicial volumes for all p ∈ T . Assertion (2) follows from the same argument using the density of Q in the ring R × p∈T Q p . If M, ∂M Qp < p, then every relative fundamental cycle k j=1 a j · σ j ∈ C * (M ; Q p ) with norm less than p satisfies |a j | p < p for all j ∈ {1, . . . , k} and so a j ∈ Z p .
Suppose that M is closed, dim M is even and that M Qp < 2p. We claim that every fundamental cycle c = k j=1 a j · σ j ∈ C * (M ; Q p ) with |c| 1,p < 2p lies in C * (M ; Z p ). Indeed, suppose that, say, a 1 ∈ Z p , then |a 1 | p = p and |a j | p ≤ 1 for all j > 1. We multiply c with p to observe that the simplex σ 1 is a cycle modulo p; this is impossible, since an even-dimensional simplex has an odd number of faces, which are summed up with alternating signs.
For all primes p, we have M, ∂M Qp ≤ M, ∂M Z (Corollary 2.10). Therefore, each prime p > M, ∂M Z satisfies the hypothesis of the first part.
We will continue the investigation of the relation between the Z p -version and the Q p -version with slightly different methods in Section 3.3.

2.4.
Scaling the fundamental class. The definition of simplicial volume · R with coefficients in a seminormed ring R clearly can be extended to all homology classes in singular homology H * ( · ; R) with R-coefficients. Proof. If m ∈ N >0 and c ∈ C dim M (M ; Z p ) is a relative cycle that represents p m · [M, ∂M ] Zp , then p · c represents p m+1 · [M, ∂M ] Zp and hence In addition, the chain p −m · c is a relative Q p -fundamental cycle of (M, ∂M ) and so M, ∂M Qp ≤ p m · |c| 1,p . Taking the infimum over all such c implies monotonicity of the sequence and shows that Conversely, let ε ∈ R >0 and let c ∈ Z(M, ∂M ; Q p ) with |c| 1,p ≤ M, ∂M Qp + ε. The relative cycle c has only finitely many coefficients; hence, there exists an r ∈ N such that all coefficients of c lie in p −r · Z p . Thus, for all m ∈ N ≥r , the relative cycle p m · c represents p m · [M, ∂M ] Zp , which yields for all m ∈ N ≥r . Taking ε → 0 then proves the claim.
Proof. If deg f = 0, the assertion is obvious. Therefore, we may assume We can now take the infimum over all c.
Corollary 2.17. Let p be a prime number and let M −→ N be a continuous map between oriented compact connected manifolds of the same dimension.
Proof. This is an immediate consequence of Proposition 2.16.
Proposition 2.18. Let R be a seminormed ring and let f : M −→ N be a finite ℓ-sheeted covering map of oriented compact connected manifolds of the same dimension. Then Proof. Let c ∈ Z(N, ∂N ; R), say c = k j=1 a j · σ j . Then the transfer is a relative R-fundamental cycle of (M, ∂M ). Since τ (σ j ) is a sum of ℓ distinct singular simplices, the triangle inequality implies the claim.

Poincaré duality and homological estimates
We will now use Poincaré duality to establish Betti number estimates, we will use the semi-simplicial sets associated with fundamental cycles to study the dependence on the primes, and we derive a simple product estimate. Variations of these arguments have been used before [11, p. 301][20, Section 3.2] in related situations.
3.1. Poincaré duality. Let M be an oriented compact connected d-manifold, let R be a ring with unit and let c = k j=1 a j σ j ∈ Z(M, ∂M ; R). By Poincaré-Lefschetz duality [12, 3.43], for each n ∈ N, the cap product map is an R-isomorphism. There is an analogous duality between H d−n (M ; R) and H n (M, ∂M ; R).

3.2.
The semi-simplicial set generated by a fundamental cycle. Let M be an oriented compact connected d-manifold and let c = k j=1 a j σ j ∈ C d (M ; R) be a relative fundamental cycle in reduced form. For each n we define X n to be the set of all n-dimensional faces of the simplices σ 1 , . . . , σ k . The ordinary face maps endow X = (X n ) n∈N with the structure of a semisimplicial set, which will be called the semi-simplicial set generated by c. The semi-simplicial subset of simplices of X contained in the boundary ∂M will be denoted ∂X. There is a canonical continuous map of pairs  By Corollary 2.9, the inequality M, ∂M (Fp) ≤ M, ∂M Z holds for all primes. In particular, the weightless simplicial F p -volume is always attained on a relative cycle with at most k := M, ∂M Z simplicies.
Say d := dim M . There are only finitely many distinct isomorphism classes of pairs (X, ∂X) consisting of a d-dimensional semi-simplicial set X generated by at most k simplices of dimension d and a semi-simplicial subset ∂X; we write S d k for a set of representatives of these isomorphism classes. Let (X, ∂X) ∈ S d k . The boundary map ∂ d : C d (X, ∂X; Z) → C d−1 (X, ∂X; Z) is a linear map between free Z-modules of finite rank and, as such, has a finite number of elementary divisors. In particular, there is a cofinite set W of primes that do not divide any elementary divisor of a boundary map ∂ d of an (X, ∂X) ∈ S d k . Let p ∈ W and let Z (p) denote the localization of Z at the prime ideal (p) ⊆ Z. Let c be a relative fundamental cycle in C d (M ; F p ) that realizes the mod p simplicial volume. We will show that c lifts to a relative cyclec ∈ C d (M ; Z (p) ) supported on the same set of simplices; by Proposition 2. Proof. We proceed as in the closed case [9, Lemma 4.1]: Let d := dim M and let c ∈ Z(M, ∂M ; R), say c = k j=1 a j · σ j . By Poincaré-Lefschetz duality, the cap product map given in (3.1) is an R-isomorphism. Hence, H n (M ; R) is a subquotient of an R-module that is generated by k elements. Therefore, rk R H n (M ; R) ≤ k. Moreover, the condition on | · | R implies that k ≤ |c| 1,R . Taking the infimum over all c gives the desired estimate. Proof. The first inequality follows from Proposition 3.2, the second inequality is contained in Corollary 2.10.
Here is a refined p-torsion estimate of the same spirit: Proof. Let us first note that p m H n (M ; Z/p m+1 Z) indeed carries a canonical F p -vector space structure.
We now proceed as in the proof of Proposition 3.2 and pick a cycle c = and thus every homology class on the right hand side can be represented by a chain on the simplices σ 1 | [0,...,n] , . . . , σ k | [0,...,n] ; i.e., the right hand side is isomorphic to a subquotient of (Z/p m+1 Z) k . Since every subquotient of (Z/p m+1 Z) k can be generated by at most k elements, we deduce that As m tends to ∞ we apply Proposition 2.15 to complete the proof.

Maximality of the fundamental class.
Similarly to the weightless case [16, Proposition 2.6, Proposition 2.10], also in the p-adic case the fundamental class has maximal norm, which in particular leads to a basic estimate for products.
is a cycle representing α. In particular, the hypothesis on the seminorm on R implies that Taking the infimum over all c proves that α 1,R ≤ M, ∂M R . Corollary 3.7 (product estimate). Let (R, | · |) be a seminormed ring that satisfies |x| ≤ 1 for all x ∈ R, and let M and N be oriented closed connected manifolds. Then Proof In particular, Proposition 3.6 and Corollary 3.7 apply to Z p : Corollary 3.8. Let p be a prime and let M and N be oriented closed connected manifolds. Then It might be tempting to go for a duality principle between singular homology with Q p -coefficients and bounded cohomology with Q p -coefficients. However, one should be aware that the considered ℓ 1 -norm on the singular chain complex is an archimedean construction (the ℓ 1 -norm) of a nonarchimedean norm (the norm on Q p ). In this mixed situation, no suitable version of the Hahn-Banach theorem can hold.
where J and K are finite sets, the coefficients a j , b k and m are integral and where p does not divide m or the a j with j ∈ J. Then Because J = ∅ and p does not divide any of the a j with j ∈ J, we see that J contains at least two elements i, j. In particular, (1) Then M, ∂M Qp ≥ 1.
Proof. As M is non-empty and closed, there exists a map M −→ S dim M of degree 1. We can now apply the degree estimate (Corollary 2.17) and the computation for spheres (Example 4.1).    (Corollary 2.10). In addition, we claim that RP d Assume for a contradiction that RP d Q 2 < 2. Then Proposition 2.14 implies that RP d Q 2 = RP d Z 2 = 2, which yields a contradiction.

4.2.
Surfaces. Recall that Σ g denotes the oriented closed connected surface of genus g and Σ g,b denotes the surface of genus g with b ≥ 1 boundary components.
Proof of Theorem 1.4. The case Σ 0 ∼ = S 2 is already contained in Example 4.1.
We first prove the inequalities "≥". Let M be Σ g or Σ g,b and let K denote the field of fractions of R. We endow K with the trivial absolute value. Using the inequality M, ∂M (R) ≥ M, ∂M (K) from Proposition 2.8, we see that it is sufficient to establish the lower bound for the field K.
Let c = k j=1 a j σ j ∈ C 2 (M ; K) be a fundamental cycle of minimal norm, i.e., |c| 1 = k is minimal. Consider the semi-simplicial set X generated by c and its chain complex We observe that dim K C 2 (X; K) = k and we claim that the kernel of ∂ 2 is 1dimensional; i.e., it is the line spanned by c. Assume for a contradiction that dim K ker(∂ 2 ) ≥ 2. In this case the relative fundamental cycles supported on {σ 1 , . . . , σ k } form an affine subspace of dimension at least 1 in C 2 (M ; K). Using elementary linear algebra we deduce that there is a fundamental cycle supported on a proper subset of {σ 1 , . . . , σ k }, which contradicts the minimality of k.
The k different simplices in X 2 have at most 3k distinct faces. Moreover, since H 2 (M, ∂M ; K) → H 1 (∂M ; K) maps the relative fundamental class ⊕ ⊕ ⊕ Figure 1. Left: Genus 0, with a single boundary component. Right: Genus 0, with two boundary components; this relative fundamental cycle consists of two singular 2simplices.
of M to a fundamental class of ∂M , the boundary of c touches every connected component of ∂M at least once. In other words, at most 3k − b faces of the simplices in X 2 are not contained in ∂M . As c is a relative cycle, every face of X 2 that is not contained in the boundary occurs at least twice. We conclude that 2 dim K C 1 (X, ∂X; K) ≤ 3k − b.
By Lemma 3.1 we have the inequality and the following calculation completes the first part of the proof Moreover, in the pathological case of Σ 0,1 , we have Σ 0,1 (R) ≥ 1 by Proposition 3.2.
In order to show that the lower bound is sharp, we construct explicit relative fundamental cycles with the desired number of 2-simplices; this is done in Proposition 4.5 below for the integral simplicial volume. As integral simplicial volume is an upper bound for · (R) (Proposition 2.8), this suffices to complete the proof. (1) Σ g Z = 4g − 2 if g ≥ 1 and (2) Σ 0,1 Z = 1 and Σ g,b Z = 3b + 4g − 4 for all g ∈ N and all b ∈ N ≥1 with (g, b) = (0, 1).
Proof. The lower bounds follow from Theorem 1.4 and Proposition 2.8. Therefore, it suffices to establish the upper bounds: Let g, b ∈ N. In the following pictures, boundary components are dashed and holes are shaded with stripes.
• In the case g = 0 and b = 2, two 2-simplices suffice (Figure 1, right). (whose edges will be identified according to the labels) into 4g − 2 simplices with the signs and orientations indicated in Figure 5. • If g ≥ 1 and b ≥ 1, we can use the construction of Figure 6 with (b − 1) triple building blocks, where the upper part of the polygon is decomposed as in the closed higher genus case ( Figure 5). Hence, Figure 4. Genus 0, with b ∈ N >2 boundary components; there are (b − 2) triple building blocks. Figure 5. Genus at least 1, empty boundary. The dotted edges will be glued as specified by the labels. For higher genus, we just increase the number of ⊖ ⊖ ⊕⊕-blocks. Figure 6. Genus at least 1 with b ∈ N ≥1 boundary components; the shaded block consists of (b − 1) triple building blocks (containing (b − 1) boundary components). The dotted edges will be glued as specified by the labels. The upper part of the polygon is decomposed as in Figure 5.
Corollary 4.6. Let g ∈ N ≥1 and let p ∈ N be prime.
Proof. Ad 1. From Theorem 1.4 and Proposition 4.5, we obtain and thus the claimed equality. Ad 2. This follows from the first part and Proposition 2.14.
Remark 4.7 (a new computation of ordinary simplicial volume of surfaces). Let g ∈ N ≥2 . Then the arguments above show that we can prove the identity Σ g = 4g − 4 for the classical simplicial volume without hyperbolic straightening: From Proposition 4.5 we know that Σ g Z = 4g − 2 (we proved this via semi-simplicial sets of cycles, without using hyperbolic straightening).
(a) We have Σ g ≤ 4g − 4: For the sake of completeness, we recall Gromov's argument [10]. For each k ∈ N, there exists a k-sheeted covering Σ g k −→ Σ g , where g k = k ·g −k +1. Hence, we obtain (Proposition 2.16) Let m ∈ N >0 and let c = k j=1 a j σ j ∈ C 2 (Σ g ; Z) be a cycle with [c] = m · [Σ g ] Z and a 1 , . . . , a k ∈ {−1, 1} as well as |c| 1 = k. Because c is a cycle, we can find a matching of the edges (and their signs) in the simplices of c such that the associated semi-simplicial set is a two-dimensional pseudo-manifold. As no proper singularities at the vertices can occur in dimension 2, this pseudo-manifold leads to a manifold, whence a surface. In other words, there is an oriented compact surface Σ (which we may assume to be connected) and a continuous map f : Σ −→ Σ g with Smoothly approximating f and looking at the corresponding harmonic representative shows that [3, p. 264] Therefore, we obtain Taking the infimum over all such cycles c proves the estimate. Similar to the case of Betti numbers or logarithmic torsion, one might wonder for which manifolds M and which coefficients R, we have M, ∂M = M, ∂M ∞ R . This question has been studied for Z-coefficients [8,9,15,5,6,7] and to a much lesser degree for F p -coefficients [16]. However, it was, for instance, not even known whether the simplicial volume of surfaces satisfies mod p approximation. With the methods developed in the previous section, we can solve this problem for surfaces: Remark 4.8 (surfaces). The simplicial volume of surfaces satisfies integral, mod p, and p-adic approximation by the corresponding normalised simplicial volumes of finite coverings: Let g ∈ N ≥1 and let p be a prime. Then we have The first two equalities are contained in Remark 4.7. Using Proposition 4.5 and Corollary 4.6, we can (in the same way) prove the last two equalities.
Hence, by Corollary 2.10, we also have In the context of homology torsion growth, it would be interesting to determine whether M ∞ (Fp) = M ∞ Zp = M holds for all oriented closed connected aspherical 3-manifolds.
Remark 4.10 (mod p Singer conjecture). Let p be an odd prime. Avramidi, Okun, Schreve established that the F p -Singer conjecture fails (in all high enough dimensions) [1], i.e., there exist oriented closed connected aspherical d-manifolds M with residually finite fundamental group and an n = d/2 such that the associated F p -Betti number gradient in dimension n is non-zero. By Corollary 3.3, also the mod p and the p-adic simplicial volume gradients are non-zero. In particular, the stable integral simplicial volume of M is non-zero.
It would be interesting to determine whether the classical simplicial volume of M is zero or not.
Remark 4.11 (estimates for groups). The Betti number estimates from Section 3.4 give corresponding estimates between the homology gradients and the stable integral simplicial volumes over the given seminormed ring. These can be turned into homology gradient estimates for groups as follows: If G is a (discrete) group that admits a finite model X of the classifying space BG, we can embed X into a high-dimensional Euclidean space R N and then thicken the image of the embedding to a compact manifold M with boundary, which is homotopy equivalent to X, whence has the same homology (gradients) as the group G. One can then study the behaviour of simplicial volumes of M to get upper bounds for homology gradients of G.

Non-values
Analogously to the case of classical simplicial volume [13], we have: Theorem 5.1. Let p ∈ N be prime and let A ⊂ N be a set that is recursively enumerable but not recursive. Then there is no oriented closed connected manifold M whose simplicial volume M Qp equals 2 − n∈N\A 2 −n .
The same statement also holds for M Zp .
The proof is based on the following notion: A real number x ∈ R is rightcomputable if the set {a ∈ Q | x < a} is recursively enumerable [23]. For example, all algebraic numbers are (right-)computable [4, Section 6] and there are only countably many right-computable real numbers.
Proof of Theorem 5.1. The numbers in this theorem are known to be not right-computable: This can be easily derived from known properties of (right-)computable numbers and Specker sequences [22]. Therefore, this theorem is a direct consequence of the observation in Proposition 5.2 below. Moreover, in combination with Proposition 2.15 and Corollary 2.12, we obtain that a ∈ Q M Qp < a = a ∈ Q ∃ m∈N >0 p m · [M ] Z Z,|·|p < a p m = a ∈ Q ∃ m∈N >0 p m , a p m ∈ S .
Because S is recursively enumerable, this set is also recursively enumerable. Hence, M Qp is right-computable.
The same arguments also can be used to show the corresponding results for oriented compact connected manifolds with boundary.