Homological Wideness, “Class Field Theory” for Covers, and a Number Theoretical Analogue

Abstract

In a previous chapter, we have constructed a particular Riemannian covering realising a wreath product. In this chapter, we first return to that example and use class field theory for Riemannian coverings (à la Sunada) to study the behaviour of geodesic in such covers. We then relate, in the general case, homological wideness of a group G acting on a manifold M (i.e., the question whether the first homology of M contains the regular representation of G) to the existence of geodesics with certain splitting behaviour. In exact analogy to an classical argument in analytic number theory, we use the Ruelle zeta function to show the existence of infinitely many totally split geodesics for a given covering in the negative curvature case. Finally, the analogy with class field theory allows us to study an analogue of homological wideness in the theory of extensions of number fields.

In a previous chapter, we have constructed a particular Riemannian covering realising a wreath product. In this chapter, we first return to that example and use class field theory for Riemannian coverings (à la Sunada) to study the behaviour of geodesic in such covers. We then relate, in the general case, homological wideness of a group G acting on a manifold M (i.e., the question whether the first homology of M contains the regular representation of G) to the existence of geodesics with certain splitting behaviour. In exact analogy to an classical argument in analytic number theory, we use the Ruelle zeta function to show the existence of infinitely many totally split geodesics for a given covering in the negative curvature case. Finally, the analogy with class field theory allows us to study an analogue of homological wideness in the theory of extensions of number fields.

1 Abelian Class Field Theory Applied to the Cover M′→ M

We briefly revert to the setup of Chap. 7, now under the stronger assumption that the action of G on M is F _ℓ-homologically wide. In this situation, the intermediate covers $M_{g_i}^{\prime \prime }$ can be characterised in terms of the splitting behaviour of certain geodesics, as we now explain.

First of all, fix $\omega \in \operatorname {H}_1(M,{\mathbf {F}}_\ell )$ to be a cyclic vector for the G-action, so that by assumption

$$\displaystyle \begin{aligned} \operatorname{H}_1(M,{\mathbf{F}}_\ell) = {\mathbf{F}}_\ell[G] \omega \oplus U \end{aligned}$$

for some complementary F _ℓ[G]-module U. Recall also that we have chosen a set of representatives {g ₁, …, g _n} for the left cosets G∕H ₁, so $\{g_1^{-1},\dots ,g_n^{-1}\}$ is a set of representatives for the right cosets H ₁∖G (by Hall’s marriage theorem, there even exists a set {g _j} that simultaneously represents the left and right cosets, so alternatively we could use the same set of representatives, if chosen suitably.) This allows us to decompose $\operatorname {H}_1(M,{\mathbf {F}}_\ell ) = {\mathbf {F}}_\ell [G] \omega \oplus U = U' \oplus U$ as direct sum of F _ℓ[H ₁]-modules, where $U'=\bigoplus _{j=1}^n {\mathbf {F}}_\ell [H_1] g_j^{-1} \omega$ is, by the assumption of homological wideness, a free F _ℓ[H ₁]-module with basis $\omega _j:=g_j^{-1} \omega$. Thus, the quotient map to the H ₁-coinvariants on U is as described in (7.2), and since this can be identified with the map q _1∗ by Lemma 7.2.1, we find an isomorphism of F _ℓ-vector spaces $\operatorname {H}_1(M_1,{\mathbf {F}}_\ell ) = \bigoplus {\mathbf {F}}_\ell \omega _j^{\prime } \oplus U_{H_1},$ where $\omega _j^{\prime }:=q_{1*}(\omega _j)$; in particular, the $\omega _j^{\prime }$ are linearly independent. With these identifications, the map q _1∗ is given as in the following diagram.

with k _i,h ∈F _ℓ, u ∈ U. Since every Riemannian covering of M is (isomorphic to) a quotient of $\widetilde M$ by a subgroup of Γ, every abelian cover of M is (isomorphic to) a quotient of $[\Gamma ,\Gamma ]\backslash \widetilde M$, and hence Galois groups of abelian covers of M correspond to quotient groups of $\operatorname {H}_1(M) \cong \Gamma ^{\operatorname {ab}}$. The coverings $\varpi _i \colon M_{g_i}^{\prime \prime } \rightarrow M$ are abelian, and they correspond, by construction, to the surjective maps

$$\displaystyle \begin{aligned} \varphi_i \colon \operatorname{H}_1(M) \xrightarrow{\otimes {\mathbf{F}}_\ell} \operatorname{H}_1(M,{\mathbf{F}}_\ell) \xrightarrow{q_{1*}=\underline{\text{t}}} \operatorname{H}_1(M_1, {\mathbf{F}}_\ell) = \bigoplus_{j=1}^n \${\mathbf{F}}_\ell \omega_j^{\prime} \oplus U_{H_1} \rightarrow C\cong {\mathbf{F}}_\ell, \\ \$\sum_{j=1}^n k_j \omega_j^{\prime} + u \mapsto k_i \mbox{ mod } \ell \end{aligned} $$

with k _j ∈F _ℓ, u ∈ U. We let { Ω_j} denote a set of linearly independent elements of $\operatorname {H}_1(M)$ that map to {ω _j} in $\operatorname {H}_1(M, {\mathbf {F}}_\ell )$.

The analogue of abelian class field theory for manifolds (described by Sunada in [93, §5], compare [90, §4]) allows us to distinguish the different covers ϖ _i, as follows.

We consider geodesics in M (smooth closed curves in M locally of minimal length) as oriented cycles, forgetting the parametrisation. A prime geodesic is a geodesic that is not a multiple of another geodesic. Let I _M denote the free abelian group generated by the prime geodesics of M, and let $I_M \rightarrow \operatorname {H}_1(M)$ denote the map that associates to a prime geodesic the homology class of the corresponding closed loop. We denote the kernel of this map by $I_M^0$, the subgroup of elements that are homologous to zero. The map is surjective: choose any lift of an element in $\operatorname {H}_1(M) \cong \Gamma ^{\mathrm {ab}}$ to Γ, and consider the free homotopy class of free loops in M corresponding to its conjugacy class; by shortening, that free homotopy class contains a closed geodesic (E. Cartan’s theorem, Note IV in his “Leçons sur la géométrie des espaces de Riemann”; see, e.g. [35, Ch. 12, Thm. 2.2]; as is written in that reference, the result of Cartan does not require negative curvature), and that geodesic maps to the given element in $\operatorname {H}_1(M)$. We conclude that there is an isomorphism $I_M/I_M^0 \cong \operatorname {H}_1(M)$.

If $\mathfrak {p}$ is a prime geodesic on M, let $(\mathfrak {p}|\varpi _i) \in {\mathbf {F}}_\ell$ denote a generator for the (cyclic) stabiliser of any lift of $\mathfrak {p}$ to $M_{g_i}^{\prime \prime }$ (since the cover ϖ _i is abelian, this does not depend on the chosen lift: in general, the stabilisers of different lifts are conjugate). By the orbit-stabiliser theorem, the number of prime geodesics in $M_{g_i}^{\prime \prime }$ above $\mathfrak {p}$ is given by $|{\mathbf {F}}_\ell |/\langle (\mathfrak {p}|\varpi _i) \rangle$. This number is either ℓ (the prime geodesic “splits”, and $(\mathfrak {p}|\varpi _i)=0$) or 1 (the prime geodesic is “inert”, and $(\mathfrak {p}|\varpi _i) \neq 0$).

A main result in abelian class field theory for manifolds, [93, Prop. 7], says, since the cover ϖ _i is abelian, the group homomorphism I _M →F _ℓ given by $\mathfrak {p} \mapsto (\mathfrak {p}|\varpi _i)$ is surjective with kernel $I_M^0 \cdot \varpi _i(I_{M_{g_i}^{\prime \prime }})$, and we have the following commutative diagram

Since $\ker (\varphi _i)$ consists of the H ₁-orbit of all homology classes spanned by both the classes Ω_j with j ≠ i, as well as the classes in the complement U, we deduce from this diagram the following result.

Proposition 10.1.1

The prime geodesics of M that are inert in the cover $\varpi _i \colon M_{g_i}^{\prime \prime } \rightarrow M$ are precisely the prime geodesics in the H ₁ -orbit of all geodesics whose homology class lies in the one-dimensional subspace 〈 Ω _i〉 of $\operatorname {H}_1(M)$.

Since g _i represent different conjugacy classes of H ₁ in G, the subspaces 〈 Ω_i〉 of $\operatorname {H}_1(M)$ are distinct, and hence so are the covers $\varpi _i \colon M_{g_i}^{\prime \prime } \rightarrow M$ in the fiber product (7.11).

2 Homological Wideness and Geodesics

The question whether the action of G on M is Q-homologically wide may be approached by splitting it into two separate questions:

• (a) Does there exist of a prime closed geodesic on M whose G-orbit consists of |G| distinct geodesics?

• (b) Do the loops corresponding to these geodesics become homologous in $\operatorname {H}_1(M,\mathbf {Q})$?

We have no general framework to deal with the second question (notice the example of the loop separating the two tori in the connected sum T ² #T ²; it is non-trivial in the fundamental group, but becomes trivial in the first homology group, since it bounds one of the tori). However, concerning the first question, we can say the following.

Proposition 10.2.1

Suppose that M and M ₀ are negatively curved closed Riemannian manifolds and M → M ₀ is a G-Galois Riemannian covering. Then there exists infinitely many closed prime geodesics in M ₀ that lift to |G| distinct closed prime geodesics in M, that hence form one G-orbit of such geodesics.

Proof

We use the following analytical argument (similar to the analytic argument that split primes in number fields exist). Let

$$\displaystyle \begin{aligned} Z_{M_0}(s):= \prod_{\mathfrak{p}} (1-e^{-s \ell(\mathfrak{p})})^{-1} \end{aligned}$$

where $\mathfrak {p}$ runs over closed prime geodesics $\mathfrak {p}$ in M ₀ of length $\ell (\mathfrak {p})$ (i.e., the Ruelle zeta function for the geodesic flow on M).

Let h > 0 denote the volume entropy of the universal covering of M (which is also the volume entropy of M ₀). Since M is negatively curved, the geodesic flow is weak mixing Anosov, so it follows that Z _M(s) converges for Re(s) > h but has a pole at s = h [75, Prop. 9].

Since the covering $M \twoheadrightarrow M_0$ is Galois, a prime geodesic $\mathfrak {p}$ splits into $r_{\mathfrak {p}}$ distinct prime geodesics in M, which are all of the same length, say, $f_{\mathfrak {p}}$ times the length of $\mathfrak {p}$. Then $r_{\mathfrak {p}} f_{\mathfrak {p}} = |G|$ (see [93, §5] or [90, §4]).

Suppose, by contradiction, that the set S of geodesics of M ₀ that split completely into |G| distinct prime geodesics in M is finite. Then $f_{\mathfrak {p}} \geq 2$ for all $\mathfrak {p} \notin S$. We find, with $\mathfrak P$ running over the closed prime geodesics of M, and for real s > h,

$$\displaystyle \begin{aligned} Z_M(s) \$ = \prod_{\mathfrak P} \left(1-e^{-s \ell(\mathfrak P)}\right)^{-1} \leq \prod_{\mathfrak{p}} \left(1-e^{-s f_{\mathfrak{p}} \ell(\mathfrak{p})}\right)^{-r_{\mathfrak{p}}} \\ \$ \leq \left(Z_{M_0}(2s)\right)^{|G|} \prod_{\mathfrak{p} \in S} \left(1+e^{-s\ell(\mathfrak{p})}\right)^{|G|}. \end{aligned} $$

(10.1)

Now $Z_{M_0}(2s)$ converges at s = h, and hence also the right hand side of the inequality (10.1) converges at s = h; this contradicts the fact that the left hand side has a pole at s = h, and hence shows that S is infinite. □

Remark 10.2.2

The result also follows from the more general Riemannian covering version of the Chebotarev density theorem due to Parry and Pollicott [76, Theorem 3] applied to the trivial conjugacy class in G.

Kojima [57, Prop. 2] gave a different proof for the case of closed orientable hyperbolic 3-manifolds admitting a totally geodesic embedding of a Riemann surface of genus ≥ 3, using projective laminations.

3 An Analogue of Homological Wideness for Number Fields

Suppose that G acts on M with (orbifold) quotient M ₀, and denote, as usual, the fundamental group of M by Γ and that of M ₀ by Γ₀. Then, as in Remark 6.2.2, G acts by outer automorphisms on Γ, and hence it acts by automorphisms on the abelianisation Γ^ab; in this interpretation, the homology representation is given by $h \colon G \rightarrow \operatorname {Aut}(\Gamma ^{\mathrm {ab}} \otimes _{\mathbf {Z}} \mathbf {Q}).$ This has the following analogue in number theory: if K∕Q is a finite Galois extension with Galois group G, and $G_K:=\mathrm {Gal}(\overline {\mathbf {Q}}/K)$, $G_{\mathbf {Q}}:= \mathrm {Gal}(\overline {\mathbf {Q}}/\mathbf {Q})$, we have a short exact sequence 1 → G _K → G _Q → G → 1, and hence an action of G by outer automorphisms on G _K, given by conjugation by any lift of an element of G to G _Q. This induces a group representation

$$\displaystyle \begin{aligned} \mathbf{h} \colon G \rightarrow \operatorname{Aut}(G_K^{\mathrm{ab}} \otimes_{\mathbf{Z}} \mathbf{Q}). \end{aligned}$$

The analogue of Q-homological wideness in this context is the following.

Proposition 10.3.1

The representation h contains the regular representation Q[G].

Proof

There exists a prime number p that is totally split in K∕Q (which follows from Chebotarev’s theorem, or easier manipulation with zeta functions much as in the proof of Proposition 10.2.1). Let $\mathfrak p$ denote any prime ideal of K above such p. Consider the reciprocity map from class field theory

$$\displaystyle \begin{aligned} \vartheta \colon {\mathbf{A}}_{K,f}^* \rightarrow G_K^{\mathrm{ab}} \end{aligned}$$

from the finite idele group ${\mathbf {A}}_{K,f}^*$ of K, let $\pi _{\mathfrak {p}}$ denote any uniformiser in the $\mathfrak p$-completion of K, and let

$$\displaystyle \begin{aligned} F:=\vartheta((1,\dots,1,\pi_{\mathfrak{p}},1,\dots,1)) \end{aligned}$$

denote a “Frobenius” of $\mathfrak p$; then for any g ∈ G,

$$\displaystyle \begin{aligned} g(F)=\vartheta((1,\dots,1,\pi_{g(\mathfrak{p})},1,\dots,1)). \end{aligned}$$

By the assumption of total splitting, all $g(\mathfrak p)$ for g running through G and for $\mathfrak p$ a fixed prime above the given p are distinct. Since the kernel of 𝜗 is the closure of the diagonally embedded K ^∗ in ${\mathbf {A}}_{K,f}^*$, we see that {g(F): g ∈ G} are distinct commuting elements of infinite order in $G_K^{\mathrm {ab}}$, and hence F is a cyclic vector for G. □

Remark 10.3.2

Compared to the case of manifolds, in the number theory case, the group $G_K^{\mathrm {ab}} \otimes _{\mathbf {Z}} \mathbf {Q}$ is of infinite rank (and captures all ramified abelian extensions), whereas $\operatorname {H}_1(M,\mathbf {Q})$ is always of finite rank (and captures topological abelian covers). For number fields, subproblem (a) as in Remark 10.2 is answered affirmatively by a similar splitting theorem as Proposition 10.2.1 for manifolds; and problem (b)—linear combinations of geodesics becoming homologous—does not occur at all, due to the specific nature of the reciprocity map.

Open Problem

Continuing along the lines of Remark 10.3.2, in [29] it was shown that isomorphism of number fields is equivalent to topological conjugacy of associated dynamical systems built from the reciprocity map. The analogous question for manifolds becomes: associate to a manifold M the dynamical system given by the monoid generated by prime geodesics acting on $\operatorname {H}_1(M,\mathbf {Z})$, where a prime geodesic acts by adding the homology class of the corresponding closed loop; then under what conditions is isometry of two manifolds M ₁ and M ₂ equivalent to the topological conjugacy of the corresponding dynamical systems, where one additionally assumes that the identification of the prime geodesics is length-preserving? (In number fields, the analogue of the additional assumption would be that the map of prime ideals preserves the norm map, but it turns out that this is automatic in that case, given the other assumptions.)

Open Problem

Study (in examples; or find a criterion) whether and how distinct geodesics in a G-orbit become homologous.