The $p$-adic Atiyah question: integrality of $p$-adic Betti numbers

Kionke (2020) introduced $p$-adic Betti numbers $b_j^{[p]}(X;\varphi,k)$, homotopy invariants of a connected finite CW-complex $X$ equipped with a virtual pro-$p$ completion $\varphi$ of its fundamental group, taking values in the $p$-adic integers $\mathbb{Z}_p$ and identified by an approximation theorem as $p$-adic limits of ordinary Betti numbers in towers of finite sheeted covers. In every example computed so far these invariants are ordinary integers, while the companion $p$-adic torsion can take transcendental $p$-adic values; integrality is proven only when the completion group is virtually abelian, and for wedge sums of such spaces. The source poses the $p$-adic Atiyah question: characterize the pairs $(X,\varphi)$ for which every $p$-adic Betti number is an ordinary integer. The question remains open in both directions: no non-integer example and no integrality theorem beyond the virtually abelian classes are known, and no values are known for mapping tori or closed hyperbolic 3-manifolds.

Background

For a prime $p$, the field $\mathbb{Q}_p$ of $p$-adic numbers is the completion of $\mathbb{Q}$ with respect to the $p$-adic absolute value, and $\mathbb{Z}_p$ denotes its ring of integers. A profinite group is a compact totally disconnected group isomorphic to an inverse limit of finite groups; a pro-$p$ group is an inverse limit of finite $p$-groups. Let $X$ be a connected finite CW-complex with fundamental group $\Gamma$. Kionke's construction starts from a virtual pro-$p$ completion of $\Gamma$: a homomorphism $\varphi\colon\Gamma\to G$ with dense image into a finitely generated profinite group $G$ having an open pro-$p$ subgroup. For a coefficient field $k$ equal to $\mathbb{Q}$ or a finite field of characteristic $\ell\neq p$, the cohomology groups of the finite covers of $X$ corresponding to open normal subgroups $N\trianglelefteq_o G$ form a directed system; the direct limit $\bar{H}^j(X;\varphi,k)$ carries a natural smooth admissible representation of $G$ (each vector is stabilized by an open subgroup). The key technical device is a $p$-adic dimension function $p\text{-}\mathrm{dim}^G_k$ on such representations: it agrees with the ordinary $k$-dimension on finite-dimensional representations and extends continuously to a $\mathbb{Z}_p$-valued function, which is possible because irreducible smooth $k$-representations of a finitely generated pro-$p$ group have $p$-power dimensions. The $j$-th $p$-adic Betti number is $b_j^{[p]}(X;\varphi,k)=p\text{-}\mathrm{dim}^G_k\bar{H}^j(X;\varphi,k)\in\mathbb{Z}_p\cup\{\infty\}$. Kionke's Approximation Theorem shows that for every chain $G_1\supseteq G_2\supseteq\cdots$ of open normal subgroups of $G$ with trivial intersection, the ordinary Betti numbers of the finite covers converge $p$-adically: $b_j^{[p]}(X;\varphi,k)=\lim_{n\to\infty}b_j(X/\varphi^{-1}(G_n);k)$. This is a $p$-adic analogue of Lück's approximation theorem, which states that the normalized ordinary Betti numbers of such a tower of covers converge to the $L^2$-Betti numbers of $X$ — the latter being the von Neumann-dimensions of the cohomology of the $L^2$-completion of the cochain complex of the universal cover. The construction is motivated by an analogy with the Atiyah conjecture, which predicts that the $L^2$-Betti numbers of a finite CW-complex with torsion-free fundamental group are integers, and which remains open in general. In the $p$-adic setting the integrality question is genuinely two-sided: each approximating Betti number is an ordinary integer, but a $p$-adic limit of integers need not be an integer, so non-integer values are not ruled out a priori. Kionke proved that all $p$-adic Betti numbers are integers when $G$ is virtually abelian (for $X$ with finite $(j+1)$-skeleton), and for wedge sums of finite connected CW-complexes with virtually abelian fundamental groups. The computed examples — tori (where the values agree with ordinary Betti numbers), surfaces (where the values do not depend on the genus), free groups (where they do not depend on rank), and infinite cyclic covers of knot complements — all give integers; by contrast, the companion invariant $t_j^{[p]}(X;\varphi,R)$, the $p$-adic torsion, can take transcendental $p$-adic values, so integrality is a distinctive feature of the Betti-number side. Integrality also has a structural consequence: when $b_j^{[p]}(X;\varphi,k)\in\mathbb{Z}$, the ordinary Betti numbers in towers such as those given by principal congruence subgroups of a $p$-adic analytic image either stabilize at the $p$-adic value or grow at least on the order of $|\Gamma:\Gamma_n|^{1/d}$ where $d$ is the dimension of the analytic group — a dichotomy relevant to cohomology growth, in the spirit of results of Bergeron–Linnell–Lück–Sauer. No $p$-adic Betti numbers are currently known for mapping tori — the space obtained from $Y\times[0,1]$ by gluing $Y\times\{0\}$ to $Y\times\{1\}$ through a self-homeomorphism of $Y$ — nor, via virtual fibering (the theorem that every closed hyperbolic 3-manifold has a finite cover fibering over the circle, so that such covers are mapping tori), for closed hyperbolic 3-manifolds; the source records these computations among the directions it leaves open.

Problem Statement

Let $p$ be a prime and let $k$ be either $\mathbb{Q}$ or a finite field of characteristic $\ell\neq p$. Let $X$ be a connected finite CW-complex with fundamental group $\Gamma$ and let $\varphi\colon\Gamma\to G$ be a virtual pro-$p$ completion of $\Gamma$, i.e. a homomorphism with dense image into a finitely generated virtually pro-$p$ group $G$. The $p$-adic Atiyah question of Kionke (2020) asks: under which conditions on $X$ and $\varphi$ are all $p$-adic Betti numbers $b_j^{[p]}(X;\varphi,k)$ ordinary integers? In full generality, the target is a characterization of the pairs $(X,\varphi)$ for which every $b_j^{[p]}(X;\varphi,k)$ lies in $\mathbb{Z}\subset\mathbb{Z}_p$ (for all degrees $j$ and all admissible coefficient fields $k$). The state of knowledge recorded in the source is: integrality holds when $G$ is virtually abelian and for wedge sums of spaces with virtually abelian fundamental groups; every example computed so far gives integer values; no example with a finite non-integer value is known; and no conjectural answer is on record, so the question is open in both directions.

References

1. Steffen Kionke, On $p$-adic limits of topological invariants, Journal of the London Mathematical Society (2) 102 (2020), no. 2, 498–534. DOI: 10.1112/jlms.12326, https://doi.org/10.1112/jlms.12326; also arXiv:1811.00356, https://arxiv.org/abs/1811.00356.
2. Jun Ueki; Hyuga Yoshizaki, The $p$-adic limits of class numbers in $\mathbb{Z}_p$-towers, Mathematical Proceedings of the Cambridge Philosophical Society 179 (2025), no. 2, 299–329. DOI: 10.1017/S0305004125000337, https://doi.org/10.1017/S0305004125000337; also arXiv:2210.06182, https://arxiv.org/abs/2210.06182.
3. Hyuga Yoshizaki, The $p$-adic limits of iterated $p$-power cyclic resultants of multivariable polynomials, preprint (2025). arXiv:2503.06194, https://arxiv.org/abs/2503.06194.
4. Sohei Tateno; Jun Ueki, The Iwasawa invariants of $\mathbb{Z}_p^{\,d}$-covers of links, Journal of the London Mathematical Society (2) 111 (2025), no. 6, Paper e70183. DOI: 10.1112/jlms.70183, https://doi.org/10.1112/jlms.70183; also arXiv:2401.03258, https://arxiv.org/abs/2401.03258.
5. Alcides Buss; Luiz Felipe Garcia; Devarshi Mukherjee, Operator algebras over the $p$-adic integers, Transactions of the American Mathematical Society 378 (2025), no. 10, 7371–7427. DOI: 10.1090/tran/9474, https://doi.org/10.1090/tran/9474; also arXiv:2403.04046, https://arxiv.org/abs/2403.04046.
6. Holger Kammeyer, Lück's Approximation Theorem, Chapter 5 (pp. 87–126) of Introduction to $L^2$-invariants, Lecture Notes in Mathematics 2243, Springer (2019). DOI: 10.1007/978-3-030-28297-4_5, https://doi.org/10.1007/978-3-030-28297-4_5.