Hausdorff dimension in R-analytic profinite groups

1 Introduction

The study of Hausdorff dimension in profinite groups was initiated by Abercrombie in [1], and has attracted special attention in recent times; see, for example, [2, 3, 4, 7, 9, 13]. If G is a countably based profinite group, a (normal) filtration of G is a descending series {Gn}n∈ℕ of (normal) open subgroups of G which is a base of neighbourhoods of 1. Throughout the paper, we always assume that G is infinite. Then we can define a metric on G by

and the topology induced by d coincides with the original topology in G. The metric d defines a Hausdorff dimension function on all subsets of G, which we denote by hDim{Gn}. As shown in [4, Theorem 2.4], if the filtration {Gn}n∈ℕ is normal and H is a closed subgroup of G, then

In [4], Barnea and Shalev studied the Hausdorff dimension in p-adic analytic pro-p groups with respect to the filtration {Gpn}n∈ℕ, and proved that

for every closed subgroup H of G. Here, if G is an analytic group, dim⁡(G) denotes the dimension of G as an analytic manifold.

In this paper we study Hausdorff dimension in R-analytic profinite groups, where R is a pro-p ring whose associated graded ring is an integral domain. We recall that a pro-p ring R is a Noetherian commutative ring with maximal non-zero ideal 𝔪 such that R/𝔪 is a finite field of characteristic p, and R is complete in the 𝔪-adic topology. The requirement for the associated graded ring to be a domain is the natural setting in the theory of analytic groups over pro-p rings, see [5, Chapter 13]. Other fundamental references for analytic groups in the general case where R is not necessarily a principal ideal domain are [10] and [11].

If G is an R-analytic profinite group, then G is virtually a pro-p group, but the subgroups Gpn need not to form a filtration, since they need not be open in G. We consider instead the natural filtration induced by a standard open subgroup of G. As we will see, the Hausdorff dimension of a closed subgroup H of G is the same with respect to the natural filtrations induced by all open standard subgroups of G. This allows us to define the concept of standard Hausdorff dimension of H, denoted by hDimStd⁡(H). We get the following theorem, in the spirit of the result of Barnea and Shalev.

An R-analytic subgroup H of G is a subgroup that is also an R-analytic submanifold of G; then H is closed in G. A crucial remark is that while the converse is true in a p-adic analytic group, i.e. every closed subgroup is analytic, it need not hold for an arbitrary pro-p ring R. For example, 𝔽p⁢[[td]] is a closed subgroup of 𝔽p⁢[[t]] for every positive integer d, but it is not analytic for d>1, even if it is an 𝔽p⁢[[t]]-analytic group in its own right.

Observe that (1.1) implies that the R-analytic spectrum of G, defined by

is finite and consists of rational values. On the other hand, the spectrum of 𝔽p⁢[[t]] corresponding to all closed subgroups is the full interval [0,1], as shown in [4, Lemma 4.1]. Thus our main theorem is pointing to the fact that most closed subgroups of 𝔽p⁢[[t]] are non-analytic.

2 Preliminaries

Throughout this paper R is a pro-p ring whose associated graded ring is an integral domain. Hence R is an integral domain, and we write K for its field of fractions. If 𝔪 is the maximal ideal of R, then R/𝔪 is a finite field of characteristic p. We will denote by q the cardinality of R/𝔪. Set H⁢(n)=dimR/𝔪⁡(𝔪n/𝔪n+1) for all n∈ℕ. For large enough n, H⁢(n) coincides with a polynomial P⁢(n), called the Hilbert polynomial of R (see [6, Chapter 8, Theorem C]).

Let G be an R-analytic group. Without loss of generality, we assume that the manifold structure is given by a full atlas. A subgroup H of G is an analytic subgroup if it is also an analytic submanifold of G. We adopt Serre’s definition of submanifold [12, Section 3.11]. Every analytic subgroup is closed in G. The converse is not true in general, but every open subgroup H of G (actually, any open subset) is a submanifold when considered with the restrictions of the charts of G, and dim⁡H=dim⁡G.

An R-analytic group S with a global chart given by a homeomorphism

is called an R-standard group of level N if ϕ⁢(1)=0 and, for all j∈{1,…,d}, there exist Fj⁢(X,Y)∈R⁢[[X,Y]] (where X and Y are d-tuples of indeterminates), without constant term, such that

Here, F=(F1,…,Fd) is called the formal group law associated to S. In these circumstances, we also say that (S,ϕ) is a standard group.

By using (2.1), F defines a new group structure on (𝔪N)(d), other than its natural additive structure, and then ϕ is an isomorphism between S and ((𝔪N)(d),F). Every R-analytic group contains an open (and so analytic) R-standard subgroup S (see [5, Theorem 13.20]). As a consequence, profinite R-analytic groups are countably based.

The formal group law of a standard group satisfies the following result (see [5, pp. 331–334]).

It follows that (𝔪N+n)(d) is a subgroup of ((𝔪N)(d),F) for all n≥0. This allows us to introduce a special type of filtrations in an R-analytic group.

Observe that

which implies that 𝔪ϕn⁢S is an open subgroup of G, and also that {𝔪ϕn⁢S}n∈ℕ is a filtration of G. Actually, we have 𝔪ϕn⁢S⁢⊴⁢S (see [5, Proposition 13.22]). Here N∈ℕ is such that S is a standard group of level N.

Hausdorff dimension can be defined for any subset of a metric space, see [8, Chapter 2] for its definition and main properties. We need the following two lemmas about Hausdorff dimension in countably based profinite groups.

Let G be a countably based profinite group and let S be an open subgroup of G. If {Sn}n∈ℕ is a filtration of S, we can calculate the Hausdorff dimension of X⊆S with respect to the metric induced by {Sn}n∈ℕ in S or in G, which we denote by hDim{Sn}S⁡(X) and hDim{Sn}G⁡(X), respectively. Our next lemma shows that there is no need to make this distinction in the notation.

3 Proof of the main theorem

In this section we first prove that the Hausdorff dimension of a closed subgroup in an analytic profinite group with respect to a natural filtration is independent of the standard subgroup. Then we prove the main theorem of our paper about Hausdorff dimension of analytic subgroups.

We need the following lemma before proving our main theorem.

We are now ready to prove our main theorem.