On the growth of Betti numbers in -adic analytic towers
We study the asymptotic growth of Betti numbers in tower of finite covers and provide simple proofs of approximation results, which were previously obtained by Calegari-Emerton [6, 7], in the generality of arbitrary -adic analytic towers of covers. Further, we also obtain partial results about arbitrary pro- towers.
Key words and phrases:Asymptotic growth of Betti numbers, -adic analytic groups
2010 Mathematics Subject Classification:58G12, 55P99
1. Introduction and statement of results
This paper is mainly concerned with the asymptotic growth of Betti numbers in a tower of finite covers of a compact space associated to a chain of subgroups of the fundamental group of which gives rise to a -adic analytic group. Both Betti numbers with coefficients in and are considered. Especially the case of -coefficients received a lot of attention in recent years. We only name here the work of Calegari-Emerton [6, 7], which is motivated by the -adic Langlands program, and the work of Lackenby [21, 22] in group theory, which is connected to property and -manifold theory.
1.1. Global setup
With the exception of section 4, we retain the following setup throughout this paper. Let be a connected compact CW-complex with fundamental group . Let be a prime, let be a positive integer, and let be a homomorphism. The closure of the image of , which is denoted by , is a -adic analytic group admitting an exhausting filtration by open normal subgroups:
Set , and let be the corresponding finite cover of . Let be the cover of corresponding to the kernel of and ; note that acts properly and freely on with quotient . Our main concern is the growth of the Betti numbers
with coefficients in and as functions of .
1.2. Growth of Betti numbers in a -adic analytic tower
W. Lück proved that for each integer the sequence always converges as , and the limit equals the -th -Betti number of the action of on . In that context we obtain the following result on the rate of convergence in terms of the dimension of as a -adic analytic group. We refer to [12, Theorem 8.36 on p. 201] for equivalent characterizations of the dimension of .
Let . Then, for any integer and as tends to infinity, we have:
The novelty of Theorem 1.1 is obviously the error
term. For more general covers, this has already been studied by Sarnak and
Xue  and by Clair and Whyte  but they
obtain much weaker results, in particular their results don’t apply when
occurs in the -spectrum of .
Theorem 1.1 generalizes in the case of trivial coefficients the main theorem of Calegari-Emerton  which deals with arithmetic locally symmetric spaces. After this paper had been put on the ArXiv Frank Calegari informed us that Theorem 1.1 can be deduced from the method of . In fact both proofs rest on a theorem of Harris, see Theorem 2.1 below, but we believe that our method of proof is somewhat simpler. We refer to Section 3 for more details on the relation to the work of Calegari-Emerton.
1.3. Growth of -Betti numbers in a -adic analytic tower
Homological algebra over Iwasawa algebras and the theory of -adic analytic groups provide important tools to study the asymptotic growth of Betti numbers in a -adic analytic tower of covers. Whilst Iwasawa algebras are hidden in the proof of Theorem 1.1, they are essential even in the formulation of a corresponding result for -Betti numbers. The Iwasawa algebra of over or is the completion of the group algebra :
The Iwasawa algebra is a right and left Noetherian domain. Further, if is torsion-free, then does not contain zero divisors and its non-zero elements satisfy the Ore condition, see [16, §6]. This means that the ring of fractions is a skew field, the Ore localization of . Hence there is a notion of rank:
If is torsion-free, we define the rank of a left -module as
For general we define the rank of as
where is any uniform, hence torsion-free, subgroup, and is regarded as an -module by restriction.
Using the above rank, we define an analog of -Betti numbers in characteristic . For a CW-complex the cellular chain complex will always be denoted by . It is a consequence of the proof of Theorem 1.1 (see (2.10)) that, if you replace by in the definition below, you obtain the -Betti numbers of .
The mod -Betti numbers of the -space are defined as
where is regarded as a right -module via .
For these characteristic analogs of -Betti numbers there is an approximation result similar to Theorem 1.1:
Let . Then for any integer and as tends to infinity, we have:
In particular, the limit of the sequence exists and is equal to .
Here again Calegari informed us that Theorem 1.4 can be deduced from his joint ongoing work with Emerton on completed cohomology. In fact one key feature of their theory is to set up the right framework to determine the growth rate of (mod ) Betti numbers even if the corresponding (mod ) -Betti number vanishes. Proving unconditional results seems difficult; we nevertheless point out that when is -dimensional the main result of  implies in particular that the error term in Theorem 1.4 is the best possible in general. We also note that – in his PhD-thesis [30, Theorem 5.3.1] – Liam Wall had first constructed examples of -adic analytic towers of covers of a finite volume hyperbolic -manifold which shows that one cannot replace the error term by for some .
There exists a link complement and a sequence of -covers of such that
We don’t know of any example with being aspherical.
1.4. Beyond -adic analytic groups
The following theorem about arbitrary pro-p towers is certainly known to some experts but we could find no proof in the literature except in degree one.
Let be a field of characteristic . Let be a compact connected CW-complex with as fundamental group. Let be a residual -chain. We denote the finite cover of associated to by . Then, for any , the sequence of normalized Betti numbers with -coefficients
is monotone decreasing and converges as .
We moreover prove that the limit is an integer in many situations, see Theorem 4.3 and the remark following it.
We thank the referee for a detailed and helpful report, especially for spotting an error in a previous version of the proof of Theorem 2.1, which is now corrected.
Work on this project was supported by the Leibniz Award of W.L. granted by the DFG. N.B. is a member of the Institut Universitaire de France. P.L. was partially supported by a grant from the NSA. R.S. thanks the Mittag-Leffler institute for its hospitality during the final stage of this project and acknowledges support by grant SA 1661/3-1 of the DFG.
In the sequel we treat the cases and simultaneously. Depending on which case, we denote by either the dimension of a -vector space or the -rank of a -module, which equals the dimension of the -vector space .
As in the work of Calegari-Emerton and Emerton [6, 7] the following result of M. Harris [18, Theorem 1.10] for which corrections appear in , is crucial. Although not explicitly stated as such, a proof is also contained in the work of Farkas-Linnell . We give a complete proof blending ideas from both Farkas-Linnell’s and Harris’ papers.
Theorem 2.1 (Harris).
Let or . Let be a finitely generated -module. Then
Here denotes the completed tensor product.
Passing to a finite index subgroup of we may, and shall, assume that is uniform and torsion-free. The proof then proceeds through a sequence of reductions.
Reduction to the case of cokernels of elements in . We first show that it suffices to show the theorem for -modules of the form
with . Let be an arbitrary finitely generated -module. Since is Noetherian, is finitely presented and we can find a matrix such that
where denotes the right multiplication
with . Since the Ore localization is a skew field, by row and column reduction in one can find invertible matrices and such that is a block matrix of the form
where is the identity matrix with and the other blocks are suitable zero matrices. In other words, describes the projection onto the first coordinates . Since are invertible, we have
There are nonzero such that are matrices over . We have
Let , and be the mod reductions of , and . Because of one obtains . Because of we have . Therefore: . Assuming the theorem is proved for modules as in (2.2), this implies that
To prove the assertion for , under the assumption that it holds for modules as in (2.2), it remains to show that
Let be the matrix such that
Let be the matrix . The same argument as before leading to (2) but now applied to shows
We have , yielding and
Note that we used here. As is invertible this implies and . In particular, and hence . So . We compute
Now (2.6) follows from (2.7).
Reduction to the case . To prove the statement for a finitely generated -module we may assume that due to the reduction to the case (2.2) and the fact that has no zero-divisors. For the following reason we may, in addition, assume that has no -torsion: Let be its -torsion part. Obviously, is a -submodule of . One easily sees by additivity of dimension that
Hence we may assume that has no -torsion. We prove now that
hence both are zero. Since the ring has finite projective dimension [3, Section 5.1] and every projective -module is free [32, Corollary 7.5.4 on p. 127] and is Noetherian, the finitely generated -module possesses a finite resolution by finitely generated free -modules:
Applying the functor yields a resolution of by finitely generated, free -modules since has no -torsion:
Now equation (2.8) follows since the rank functions over and are additive and the equation obviously holds for finitely generated free -modules.
Let . Because of it is enough to prove . This follows from the -case, , and the inequality
So we reduced the proof of the theorem to the case and henceforth assume .
Reduction to being standard. We finally reduce the assertion to the case that is standard in the sense of [12, §8.4]. Being a -adic analytic group, has an open subgroup which is standard with respect to the manifold structure induced from , see [12, Theorem 8.29]. Since is open, we have for greater than some . Being standard has a preferred collection of open normal subgroups which satisfy: (); see e.g., [12, Ex. 6 p. 168].
Recall that we may assume that due to the reduction to the case (2.2) and the fact that has no zero-divisors.
Now if the assertion holds for with respect to the ’s, then it follows that for the
left hand side of (2.1) is bounded by a constant times
and is therefore ,
so the assertion holds for as well. We assume from now on that is standard
and that where is the global atlas of .
The remaining argument. Let be as in (2.2). We may assume . By a fundamental result of Lazard, the graded ring with respect to the filtration by powers of the augmentation ideal is a polynomial algebra with indeterminates [32, Theorem 8.7.10 on p. 160], where is a minimal generating set. Let be the closure of the ideal generated by elements with . Note that for any -module . Since is a domain, . Now for each integer (if ) or (if ), the global atlas of induces an epimorphism with kernel . It therefore follows that for some rational constant and we have to show that
But it follows from [12, Lemma 7.1] that there exists a positive integer such that for all . It therefore suffices to show (2.9) with replaced by . Let be such that . Let be the map induced by right multiplication with . We have
The last equality follows from the fact the graded ring is a polynomial ring. For the same reason the last number equals the number of monomials in a polynomial ring with variables each of which has total degree in the interval . The number of monomials of degree is . Hence
As a polynomial in , each binomial coefficient has leading term . Their difference is a polynomial in with degree at most . This implies (2.9). ∎
We show for both cases and simultaneously that
The CW-structure on lifts to a -equivariant CW-structure on and to -equivariant CW-structures on . We may also view as a -space via the quotient map . Let be the cellular chain complex of . Each chain module is a finitely generated free -module. The differentials in the chain complex are denoted by . Note that is isomorphic to the cellular chain complex as an -chain complex. In particular, we have
We write and short for and its differentials. We denote the cycles and boundaries in the chain complexes and by , and , , respectively. Let be the rank of the finitely generated free -module . In each degree we have the obvious exact sequence
By additivity of we obtain that
By (2.11), a similar argument as above, and right-exactness of the tensor product, we obtain that
The natural map
induced by is a right -module isomorphism (recall that we regard as a right -module via ). The inverse is obtained as follows: Since , there is a natural continuous homomorphism from to the invertible elements of the -algebra . By the universal property of the completed group algebra there is a continuous homomorphism which descends to the desired inverse. As a consequence we get isomorphisms
Now (2.10) follows from (2), (2.14), and Theorem 2.1. Note that (2.10) is exactly the statement of Theorem 1.4 in the case . Next we explain how Theorem 1.1 follows from (2.10) when . Since has characteristic zero, we have . Since as , we conclude
3. Relation with the completed homology
These modules carry continuous actions of and may therefore be considered as -modules or -modules, respectively. In this section we want to clarify the relation of completed cohomology to (mod ) -Betti numbers.
Retaining the setup in section 1.1 we have:
Here again we may reduce to the case where is torsion-free. Write . The claim is equivalent to
having the same -rank. So the statement is equivalent to:
Since is a tower of chain complexes of abelian groups satisfying the Mittag-Leffler condition, by [31, Theorem 3.5.8] there is a short exact sequence
Moreover, since towers of finite dimensional vector spaces over a field satisfy the Mittag-Leffler condition, we conclude that
which yields the proposition. ∎
It follows from works of Calegari and Emerton that a similar result with