Minimal Geodesics and Nilpotent Fundamental Groups

Bernd Ammann

Mathematisches Institut der Universität Freiburg D-79104 Freiburg Germany E-Mail: ammann@arcade.mathematik.uni-freiburg.de

March 1996

Abstract. Hedlund [18] constructed Riemannian metrics on n-tori, n ≥ 3 for which minimal geodesics are very rare. In this paper we construct similar examples for every nilpotent fundamental group. These examples show that Bangert’s existence results of minimal geodesics [4] are optimal for nilpotent fundamental groups.

AMS Subject Classification (1991): Primary 53C22, Secondary 22E25, 20F18:

Key words: minimal geodesics, stable norm, first Betti number, nilpotent Lie groups, cocompact discrete subgroups, nilmanifolds, Hedlund metrics

1. Introduction

DEFINITION 1.1. Let (M, ⟨ . , . ⟩) be a Riemannian manifold. A nonconstant geodesic c: ℝ → M is called minimal if it satisfies for all t_{1} < t_{2}:

L (c|_{[t_{1},t_{2}]}) ≤ L (γ)

for all curves γ homotopic to c|_{[t_{1},t_{2}]} with fixed endpoints.

Suppose we fix π_{1}(M). Then there are several known results that guarantee the existence of minimal geodesics.

The simplest one is that (M, ⟨ . , . ⟩) carries a minimal geodesic if and only if π_{1}(M) is infinite.

For some classes of differentiable manifolds certain existence properties of minimal geodesics do not depend on the choice of Riemannian metric: the bestknown cases are compact manifolds M with hyperbolic fundamental groups. Here one can compactify the universal cover M̃ of M by a “boundary at infinity” M̃_{∞}. For every Riemannian metric on M the lift of a minimal geodesic to M̃ converges for t → ±∞ to two different points on M̃_{∞} and, conversely, for each pair of different points on M̃_{∞} there exists such a minimal geodesic ([23],[8],[19],[17]7.5).

The situation is similar on the 2-torus T^{2} = ℝ^{2}/ℤ^{2} where for every straight line in ℝ^{2} and every Riemannian metric on T^{2} one finds a minimal geodesic whose lift stays at finite distance from the straight line ([18],[7],[3]).

Surprisingly, the situation is completely different for an n-torus T^{n} = ℝ^{n}/ℤ^{n} if n ≥ 3. Here existence properties of minimal geodesics depend very much on the choice of the Riemannian metric: for flat metrics every geodesic is minimal (and lifts to a straight line in ℝ^{n}). On the other hand, there are the Hedlund metrics [18] on T^{n}, discussed in [4], where one has only n periodic minimal ones. So in these Hedlund examples minimal geodesics are very rare. Using the language of dynamical systems one would say that the set of unit tangent vectors to minimal geodesics consists of 2n periodic orbits of the geodesic flow and (countably many) heteroclinic and homoclinic connections between them.

These Hedlund metrics contrast to a theorem of V. Bangert ([4],[5]). He proves that the number of “directions” of minimal geodesics on an arbitrary Riemannian manifold (M, ⟨ . , . ⟩) is at least the first Bettinumber b_{1} := rank π_{1}/[π_{1}, π_{1}]. This bound is optimal for the Hedlund metric on T^{n}, n ≥ 3.

In this paper we will construct Riemannian manifolds with only b_{1} different “directions” of minimal geodesics for arbitrary nilpotent fundamental groups. Therefore Bangert’s bound is optimal for arbitrary nilpotent groups.

If one tries to construct such Riemannian manifolds using analogous methods to Hedlund’s, one has to prove a group theoretical property for the fundamental group. The groups having this property will be called groups of bounded minimal generation. Any finitely generated abelian group is of bounded minimal generation, and every group of bounded minimal generation is virtually nilpotent, i.e. it has a nilpotent subgroup of finite index. Unfortunately, there are only few non abelian groups that are known to be of bounded minimal generation, e.g. discrete subgroups of Heisenberg groups (see section 7). So this type of construction seems to fail for general fundamental groups.

Therefore we will use a different method that will give us examples for any finitely generated nilpotent fundamental group.

The Riemannian manifolds we construct have a universal covering M̃ = G × S̃ where G is a nilpotent Lie-group and S̃ is a simplyconnected compact manifold. The commutator group [G, G] acts isometrically on M̃ via left multiplication on the first component. In analogy to the Hedlund metrics on tori we will find two types of minimal geodesics on M: the left-translated-periodic type and the connection type. There are b_{1} periodic minimal geodesics c_{1}, …, c_{b_{1}} on M with lifts c̃_{1}, …, c̃_{b_{1}} on M̃ with the following property: for every minimal geodesic c of left-translated-periodic type we can find a lift c̃ to M̃ and i ∈ {1, …, b_{1}}, a, b ∈ ℝ, g ∈ [G, G] with

c̃(t) = g · c̃_{i}(at + b).

On the other hand every c: ℝ → M satisfying this property is a minimal geodesic.

A minimal geodesic of connection type will always be a homoclinic or heteroclinic connection between two geodesics of left-translated-periodic type.

Additionally, the main theorem for this construction (Theorem 2.1) is useful for other applications. For example, we will be able to determine all minimal geodesics on nilmanifolds with left-invariant metrics. Here minimal geodesics are exactly the horizontal lifts of straight lines on the associated (flat) Jacobi variety T^{b_{1}}. Acknowledgement This paper resulted from the author’s Diplomarbeit at the University of Freiburg, Germany. The author wants to thank his supervisor Victor Bangert for many interesting discussions.

2. N-leftinvariant metrics

In this section we will look at metrics on G × S̃ as above. Applying theorem 2.1, we will be able to reduce the classification problem of minimal geodesic on G × S̃ to the classification of minimal geodesics on ℝ^{b_{1}} × S̃, or to be more precise:

the minimal geodesics on G × S̃ are exactly the horizontal lifts of minimal geodesics on ℝ^{b_{1}} × S̃ via the canonical Riemannian submersion

G × S̃ → ℝ^{b_{1}} × S̃.

We will formulate the theorem in a more general setting.

Let G be a simply connected, nilpotent Lie-group. The Lie-group exponential map exp is a global diffeomorphism from the Lie-algebra � to the Lie-group G, and the Formula of Baker, Campbell and Hausdorff states that the pullback of the multiplication on G is a Lie-bracket polynomial on �. Connected subgroups of G correspond to Lie-subalgebras of � and are therefore closed subsets. Normal connected subgroups correspond to ideals of �. For details and further results about nilpotent Lie-groups look for example at [10] and [25].

We fix now a normal connected subgroup N of G with Lie-algebra � ⊂ �. We will assume that N is contained in the commutator group [G, G] of G. This is equivalent to the condition that � is in the commutator Lie-algebra [�, �] of �.

Let S be a compact manifold; here we do not assume that S is simply connected but we want π_{1}(S) to be finite. G acts on G × S via left multiplication on the first component. Now we take a Riemannian metric ⟨ . , . ⟩_{G} on G×S that is N-leftinvariant, i.e. the subgoup N of G acts isometrically. Then there is a unique Riemannian metric ⟨ . , . ⟩_{G/N} on (G/N)×S such that the canonical projection p: G×S → (G/N)×S is a Riemannian submersion. Vice versa, for every Riemannian metric on (G/N) × S there is a (non unique) N-leftinvariant metric on G × S such that p is Riemannian.

Additionally we suppose that ⟨ . , . ⟩_{G} is bi-Lipschitz to a left invariant metric ⟨ . , . ⟩_{l} on G × S, i.e. there are constants c_{1}, c_{2} > 0 with

c_{1}⟨v, v⟩_{l} ≤ ⟨v, v⟩_{G} ≤ c_{2}⟨v, v⟩_{l} ∀v ∈ T (G × S).

This condition is independent of the choice of the left invariant metric ⟨ . , . ⟩_{l}.

A vector v ∈ T_{x}(G × S) is called horizontal if v ⊥ ker T_{x}p.

THEOREM 2.1. Let N be a normal connected subgroup of the simply connected, nilpotent Lie-group G with N ⊂ [G, G]. We suppose that G × S carries a Riemannian metric that is N-left-invariant and bi-Lipschitz to a G-left-invariant metric and that

p: G × S → \frac{G}{N} × S

is a Riemannian submersion. Then c: ℝ → G×S is a minimal geodesic on (G × S, ⟨ . , . ⟩_{G}) if and only if

1. ċ(t) is horizontal for all t ∈ ℝ and

2. p∘c: ℝ → (G/N)×S is a minimal geodesic on ((G/N) × S, ⟨ . , . ⟩_{G/N} ).

Proof. Let Z_{1}(G) := {x ∈ G | xyx^{−1}y^{−1} = e ∀y ∈ G} be the center of G and define inductively Z_{i+1}(G) := {x ∈ G | xyx^{−1}y^{−1} ∈ Z_{i}(G) ∀y ∈ G}.

We will prove the theorem for the case N ⊂ Z_{1}(G). By a straightforward induction on i we then get the theorem for N ⊂ Z_{i}(G) and therefore the general case.

To prove “⇐” we suppose that c is not minimal. If ċ(t) is horizontal for all t, then c|_{[s,t]} has the same length as p ∘ c|_{[s,t]} and therefore p ∘ c cannot be minimal.

p ∘ c(0)  p ∘ c(t)^{c(0) = ĉ(0)  ĉ(t) c(t)} fibres of G × S → \frac{G}{N} × S  p \frac{G}{N} × S G × S

Figure 1. Minimal Geodesics are horizontal

             minimal.tex; 4/07/1997; 15:44; no v.; p.5

For “⇒” we suppose that c: ℝ → G × S is a minimal geodesic, parametrized by arclength. Without loss of generality we can assume that S is simply connected.

For any v ∈ � let r_{exp v} be the right-translation of G × S by exp v acting trivially on S. Then V_{v} := \frac{∂}{∂α}|_{α=0}r_{exp αv} is a left-invariant vector field with vanishing S-component.

If n ∈ � then r_{exp αn} acts isometrically on G × S, since N ⊂ Z_{1}(G), so V_{n} is a Killing field. Noether’s theorem ([2] 4.20) implies that

P_{n} := ⟨ċ(t), V_{n}(c(t))⟩_{G}

is constant in t. We argue by contradiction to show that P_{n} = 0.

We write

ċ(t) = λ(t)V_{n}(c(t)) + c_{⊥}(t)

with c_{⊥}(t) ⊥ V_{n}(c(t)).

Let ∥ . ∥_{G} be the norm of tangential vectors induced by ⟨ . , . ⟩_{G}. If we assume that P_{n} ≠ 0 we can use the Lipschitz constants between ⟨ . , . ⟩_{G} and a left-invariant metric to obtain constants K_{1}, K_{2} > 0 in the inequalities:

|λ(t)| = \frac{|⟨ċ(t), V_{n}(c(t))⟩_{G}|}{∥V_{n}(c(t))∥_{G}^{2}} = \frac{|P_{n}|}{∥V_{n}(c(t))∥_{G}^{2}} < K_{1}

∥λ(t)V_{n}(c(t))∥_{G} = \frac{|P_{n}|}{∥V_{n}((c(t))∥_{G}} > K_{2} > 0

Then the curve ĉ defined by

ĉ(t) := c(t) · exp (( ∫_{0}^{t} −λ(t′)dt′)n)

satisfies p∘ĉ = p∘c, ĉ(0) = c(0) and ˙ ĉ(t) ⊥ V_{n} (see also Figure 1). After identification of T_{ĉ(t)}M and T_{c(t)}M via left translation, ˙ ĉ(t) is equal to c_{⊥}(t).

So we know that ∥˙ĉ∥_{G} ≤ \sqrt{1 − K_{2}^{2}}. Writing d_{G} for the distance induced by ⟨ . , . ⟩_{G} we obtain

t = d_{G}(c(0), c(t)) ≤ \underbrace{d_{G}(ĉ(0), ĉ(t))}_{≤(\sqrt{1−K_{2}^{2}})t} + d_{G}(ĉ(t), c(t)) ∀t > 0. (1)

We use a result of Pansu ([24]) to state that there is a constant K_{3}(n) not depending on α > 0, s ∈ S and g ∈ G such that

d_{G}((g exp αn, s), (g, s)) ≤ K_{3}(n) (\sqrt{α} + 1).  (2)

Pansu did not prove exactly this statement, but the proof of it is completely analogous to the proof of [24] no. (62) if we use the fact that exp αn is in the commutator group. Another proof using more elementary methods can be found in [1].

Together with |λ(t)| < K_{1} inequality (2) contradicts (1), so we get P_{n} = 0 for every n in the Lie algebra of N, i.e. ċ(t) is horizontal. This implies that p ∘ c is parametrized by arclength.

It remains to show that p ∘ c is minimal. In order to prove it we assume the opposite, i.e.

Δ := t_{2} − t_{1} − d_{G/N}(p ∘ c(t_{1}), p ∘ c(t_{2})) > 0.

Now take a shortest geodesic k̄: [t_{1}, t_{2}] → (G/N) × S from p ∘ c(t_{1}) to p ∘ c(t_{2}) (see also Figure 2). This shortest geodesic has a unique horizontal lift k: [t_{1}, t_{2}] → G × S with c(t_{1}) = k(t_{1}).

As the Lie exponential map is a diffeomorphism there is a unique n ∈ �, such that k(t_{2}) = c(t_{2}) · exp n. For μ > 0 we now extend k continuously by

k(t) := c(t) · exp([1 − μ(t − t_{2})]n) t_{2} ≤ t ≤ t_{μ} := t_{2} + 1/μ.

So k: [t_{1}, t_{μ}] → G × S is also a curve from c(t_{1}) to c(t_{μ}). We will prove that k is shorter than c|_{[t_{1},t_{μ}]} for small μ > 0. There is a unique c′ : ℝ → �^{⊥} ⊂ � such that the G-component of ċ(t) is V_{c′(t)}(c(t)). On (t_{2}, t_{μ}) the G-component of k̇(t) is

V_{c′(t)}(k(t)) − μV_{n}(k(t)),

whereas the S-components of k̇(t) and ċ(t) are equal up to left (or right) translation.

So as c is horizontal

∥k̇(t)∥_{G} = \sqrt{∥ċ(t)∥_{G}^{2} + μ^{2}∥V_{n}(k(t))∥_{G}^{2}}
≤ 1 + \frac{1}{2}μ^{2}∥V_{n}(k(t))∥_{G}^{2}.

For μ small enough we get

L(c|_{[t_{1},t_{μ}]}) − L(k) ≥ Δ − \frac{1}{μ} \frac{μ^{2}}{2} sup_{{g∈G \atop s∈S}} ∥V_{n}(g, s)∥_{G}^{2} > 0,

which contradicts the minimality of c. Therefore p ∘ c is minimal. □

Using Theorem 2.1 we now know the minimal geodesics on any nilpotent Lie-group with a left-invariant metric.

p ∘ c(t_{μ}) p ∘ c(t_{1})  p ∘ c(t_{2}) p ∘ c k̄ = p ∘ k^{c(0) = k(0)  c(t_{μ}) = k(t_{μ}) c(t_{2}) k(t_{2}) c k} p \frac{G}{N} × S G × S

Figure 2. Minimal Geodesics project to Minimal Geodesics

                    minimal.tex; 4/07/1997; 15:44; no v.; p.8

COROLLARY 2.2. Let G be a nilpotent Lie-group with a left-invariant metric. The minimal geodesics on G are exactly the curves of the form

c(t) = g · exp tv

with g ∈ G and v ∈ � and v ⊥ [�, �].

Remark. A similar type of orthogonality relation was discovered by Patrick Eberlein, Ruth Gornet and Dorothee Schüth when they investigated the following problem. A geodesic c on a nilpotent Lie-group G with left-invariant metric is called periodic if there are g ∈ G, λ > 0 with c(t + λ) = g · c(t) ∀t. A necessary condition for periodicity is that periodic geodesics are orthogonal to certain terms built by commutators ([11]Cor. 4.4, [14]3.1).

Now we try to lift asymptotic behavior from (G/N) × S to G × S. Here we have to pay attention to the following fact: if dim N > 0 then there are curves γ̄_{1}, γ̄_{2}: ℝ → (G/N) × S that are asymptotic to each other but do not have horizontal lifts that are asymptotic to each other.

The situation is different if we replace “asymptotic” by “exponentially asymptotic”.

DEFINITION 2.3. The (parametrized) curves γ_{1} and γ_{2} are exponentially asymptotic for t → ∞ if there are constants K_{a}, K_{b} > 0 such that

d (γ_{1}(t), γ_{2}(t)) < K_{a}e^{−K_{b}t} for large t.

Here d is the Riemannian distance. This definition is invariant under bi-Lipschitz change of the metric. The definition of exponentially asymptotic for t → −∞ is analogous.

PROPOSITION 2.4. Let G × S carry a Riemannian metric that is Nleft-invariant and invariant under left-action of a lattice Γ of G. Choose a Riemannian metric on the quotient such that p : G × S → (G/N) × S becomes a Riemannian submersion. Furthermore let γ_{1}, γ_{2}: ℝ → G × S be piecewise C^{1}-curves with bounded ∥γ̇_{i}∥_{G} and horizontal with respect to p. Then p ∘ γ_{1} is exponentially asymptotic to p ∘ γ_{2} if and only if there is an n_{∞} ∈ N such that n_{∞} · γ_{1} is exponentially asymptotic to γ_{2}.

Proof. We only have to prove the “only if”. And it is sufficient to prove this for N ⊂ Z_{1}(G) as the general case follows by induction. The cases t → +∞ and t → −∞ are totally symmetric, so the case t → −∞ will be omitted.

Because of the action of Γ and the compactness of S the injectivity radius of G/N × S is positive.

At first we can choose t_{0} with

d(p ∘ γ_{1}(t), p ∘ γ_{2}(t)) < \frac{1}{2} inj rad(\frac{G}{N} × S) ∀t ≥ t_{0}.

We glue a surface A: [1, 2]×[t_{0}, ∞) → (G/N)×S between the p ∘ γ_{i}|_{[t_{0},∞]} such that A(i, .) = p ∘ γ_{i} (i = 1, 2) and A(., t) is the shortest curve from p ∘ γ_{1}(t) to p ∘ γ_{2}(t).

As p ∘ γ_{1}(t) and p ∘ γ_{2}(t) are exponentially asymptotic, there are constants K_{a}, K_{b} > 0 with

Area A|_{[1,2]×[t_{1},t_{2}]} < K_{a}e^{−K_{b}t_{1}} (t_{0} ≤ t_{1} ≤ t_{2} ≤ ∞).

Now lift A to Ã: [1, 2] × [t_{0}, ∞) → G × S such that Ã(1, .) = γ_{1} and Ã(., t) is horizontal ∀t. There is an n: [t_{0}, ∞) → N with

Ã(2, t) · n(t) = γ_{2}(t).

In general n is non-constant and therefore Ã(2, .) is non-horizontal, but we will show that n(t) converges for t → ∞.

Note that G × S → (G/N) × S is a principal N-bundle. As N acts isometrically, the horizontal planes determine a connection-1-form ω: T (G × S) → �. (For details on connection-1-forms see [20], Chapter II.) Then dω is the curvature of the connection and ∥dω∥ is uniformly bounded on G × S.

As γ_{i} and Ã(., t) are horizontal

∫_{_{γ_{i}}|_{[t_{1},t_{2}]}} ω = 0 and ∫_{Ã(.,t)} ω = 0

∫_{∂(Ã|[1,2]×[t_{1},t_{2}])} ω = ∫_{_{Ã(2,.)}|_{[t_{1},t_{2}]}} ω = n(t_{1})n(t_{2})^{−1}

Using Stoke’s Theorem we get

d (n(t_{1})n(t_{2})^{−1}, e) ≤ ∫_{Ã|_{[1,2]×[t_{1},t_{2}]}} ∥dω∥ ≤ Area (A|_{[1,2]×[t_{1},t_{2}]}) sup_{G×S} ∥dω∥
≤ sup ∥dω∥K_{a}e^{−K_{b}t_{1}}.

So n(t) converges exponentially and n_{∞} := lim_{t→∞} n(t) gives the proposition. □

3. Hedlund examples

In this chapter we will give a slight generalisation of the Hedlund examples presented by Bangert in [4], section 5. The proofs are only small variations of Bangert’s proofs, so we will skip them.

In this section we construct similar metrics, which we will also call “Hedlund metrics”. These metrics are defined on manifolds of the form M = T^{b_{1}} × S where T^{b_{1}} = ℝ^{b_{1}}/ℤ^{b_{1}}, b_{1} ≥ 1 is the torus and S is an arbitrary compact connected manifold with finite π_{1}(S). We exclude the case M = T^{2} by assuming dim S > 0 or b_{1} ≠ 2. Note that b_{1} is the first Betti number of M.

We denote the standard flat metric on T^{b_{1}} by ⟨ . , . ⟩_{T} and we choose a metric ⟨ . , . ⟩_{S} on S. The product metric on M will be called ⟨ . , . ⟩_{T ×S}. The vectors of the canonical basis e_{1}, …, e_{b_{1}} of ℝ^{b_{1}} induce ⟨ . , . ⟩_{T ×S}- orthonormal vector fields E_{1}, …, E_{b_{1}}: M → T M.

The Hedlund metrics ⟨ . , . ⟩_{H} will be defined in Definition 3.1. In this definition we use b_{1} closed curves c_{1}, …, c_{b_{1}} on M that will become the only geodesics that are minimal and closed. To define them, we have to distinguish two cases.

In the case “b_{1} ≠ 2” choose s ∈ S and define

c_{i}(t) := ( \frac{\frac{1}{2}e_{i} + te_{i+1}}{ℤ^{b_{1}}} , s )

for all t ∈ ℝ, where e_{b_{1}+1} := e_{1}.

In the case “b_{1} = 2” we have assumed that dim S > 0, so we can choose different s_{1}, s_{2} ∈ S and define

c_{i}(t) := (\frac{te_{i}}{ℤ^{b_{1}}} , s_{i} )

for all t ∈ ℝ.

In both cases let L_{i} be the trace of c_{i}. The fact that L_{i} and L_{j} are disjoint for i ≠ j plays an important role in the proofs. The construction of Hedlund type metrics on T^{2} fails because such c_{i} and L_{i} do not exist on T^{2}.

Now define U_{ϵ}(L_{i}) to be the ϵ-neighborhood of L_{i} with respect to ⟨ . , . ⟩_{T ×S}.

For ϵ > 0 (that will be chosen very small) we define in analogy to Definition 5.1 of [4]:

DEFINITION 3.1. ⟨ . , . ⟩_{H} is an ϵ-Hedlund metric on M iff there are ϵ_{1}, …, ϵ_{b_{1}} ∈ (0, ϵ] such that for i = 1, …, b_{1}:

(P 1) ⟨v, v⟩_{H} ≤ (1 + ϵ)^{2}⟨v, v⟩_{T ×S} ∀v ∈ T M
(P 2) ⟨E_{i}(x), E_{i}(x)⟩_{H} = ϵ_{i}^{2}  ∀x ∈ L_{i}
⟨v, v⟩_{H} ≥ ϵ_{i}^{2}⟨v, v⟩_{T ×S}  ∀v ∈ T_{x}M, x ∈ L_{i}
⟨v, v⟩_{H} > ϵ_{i}^{2}⟨v, v⟩_{T ×S}  ∀v ∈ T_{x}M \ {0}, x ∈ U_{ϵ}(L_{i}) \ L_{i}
(P 3) ⟨v, v⟩_{H} ≥ ⟨v, v⟩_{T ×S}  ∀v ∈ T_{x}M, x ∉ ⋃_{j} U_{ϵ}(L_{j})

The following propositions 3.2, 3.4 and 3.5 are analogues to Propositon 5.2, Proposition 5.3 and Corollary 5.4 of [4]. Because of the definition of ϵ-Hedlund metric it is clear that any statement in these propositions that holds for ϵ > 0 also holds for any ϵ′ ∈ (0, ϵ).

PROPOSITION 3.2. There is an ϵ > 0 and a K_{1} ∈ ℝ such that for any ϵ-Hedlund metric on M and any arclength-parametrized minimal geodesic c with respect to this metric the length of

A := c^{−1}(M \ ⋃_{i} U_{ϵ}(L_{i})) ⊂ ℝ

is bounded by K_{1}.

That means that c “stays out of ⋃_{i} U_{ϵ}(L_{i}) only for a bounded time”. As an immediate consequence c cannot change its “tube” too often. To make this precise we define:

DEFINITION 3.3. Let ϵ > 0 be so small that the U_{ϵ}(L_{i}) (i = 1, …, b_{1}) are disjoint and let c: ℝ → M be a minimal geodesic. We define the change number C(c) ∈ ℕ∪{0, ∞} to be the supremum of all n ∈ ℕ such that we find t_{0} < t_{1} < … < t_{n} and i_{j} ∈ {1, …, b_{1}} with c(t_{j}) ∈ U_{ϵ}(L_{i_{j}}) and i_{j} ≠ i_{j+1}.

PROPOSITION 3.4. There is an ϵ > 0 and K_{2} ∈ ℕ such that C(c) ≤ K_{2} for any minimal geodesic c with respect to any ϵ-Hedlund metric on M.

Remark. If S = {one point} Bangert proved in [4] that we can even find ϵ > 0 with K_{2} := b_{1}. For general S this statement does not hold.

PROPOSITION 3.5. There is an ϵ > 0 such that every minimal geodesic on an ϵ-Hedlund metric on M is asymptotic in each of its senses to one of the L_{i}’s.

In section 5 we will need a stronger version, so we formulate a supplement.

SUPPLEMENT 3.6. If the c_{i} are even hyperbolic closed geodesics, e.g. if

A_{jk}(x) := E_{k}(E_{j}(⟨E_{i}, E_{i}⟩_{H}))(x) j, k ≠ i

is positive definite for all x ∈ L_{i}, then any minimal geodesic c is exponentially asymptotic to one of the L_{i} in each of its senses (see Definition 2.3).

Remark. It is also possible to formulate analogues to the propositions 5.6, 5.7 and 5.8 from [4].

4. Lattices in nilpotent Lie-groups

Here we will summarize some facts used in the next section. For the discrete group or Lie-group G we define the descending central series (G^{i})_{i∈ℕ} inductively by G^{1} := G and G^{i+1} := [G, G^{i}]. Then G is nilpotent iff G^{i} = {e} for sufficiently big i ∈ ℕ.

THEOREM 4.1 (Malcev, [25] theorem 2.18). A group Γ is isomorphic to a lattice in a nilpotent, simply connected Lie-group iff Γ is finitely generated, nilpotent and torsion free.

THEOREM 4.2. Let Γ be a lattice in the nilpotent, simply connected Lie-group G and N a closed normal subgroup (not necessarily connected), p: G → G/N. If two of the following three conditions are true, the third follows:

1. Γ ∩ N is a lattice in N,

2. p(Γ) is a lattice in G/N,

3. Γ is a lattice in G.

“1. and 2. ⇒ 3.” and “1. and 3. ⇒ 2.” are proved in [10] lemma 5.1.4, the proof of “2. and 3. ⇒ 1.” is straightforward.

THEOREM 4.3. Let Γ be a lattice in G, then Γ^{i} is a lattice in G^{i} for i ∈ ℕ.

This follows from the theory of Malcev bases ([10]) and [10] corollary 5.4.5. It is a slight generalisation of [25] corollary 1 of theorem 2.3 saying that Γ ∩ G^{i} is cocompact in G^{i}.

Using the theory of Malcev bases it is also evident that

rank \frac{Γ}{[Γ, Γ]} = dim \frac{G}{[G, G]}.

5. Main construction

In this section we will construct our examples with minimal geodesics in only “few directions” by combining the results we obtained in the previous sections.

THEOREM 5.1. For any finitely generated nilpotent group Π_{1} we find a connected compact Riemannian manifold (M, ⟨ . , . ⟩_{M}) satisfying:

(1) π_{1}(M) = Π_{1}

(2) M has a universal covering M̃ = G × S̃ where G is a nilpotent Lie-group and S̃ is a compact manifold.

(3) The commutator group [G, G] acts isometrically on the Riemannian covering M̃ via left multiplication on the first component.

(4) There are minimal geodesics c_{i}: ℝ → M (i ∈ {1, …, b_{1}}) with lifts c̃_{i}: ℝ → M̃ such that every minimal geodesic γ: ℝ → M is of one of the following types:

Type I: left-translated-periodic γ has a lift γ̃: ℝ → M̃ such that there are a, b ∈ ℝ, g ∈ [G, G],

i ∈ {1, …, b_{1}} with γ̃(t) = g · c_{i}(at + b).

Type II: connection type γ is not of Type I, but there are minimal geodesics γ_{+} and γ_{−} of Type I such that γ(t) is exponentially asymptotic to γ_{±}(t) for t → ±∞.

Any compact manifold is a finite CW-complex and therefore the fundamental group of any compact manifold is finitely genrated.

The author has presented another version of this construction in [1] and proves that if Π_{1} has no torsion and Π_{1} ≠ ℤ^{2} we even get an example as above with M̃ = G×{one point} (but without property (3) in the case b_{1} = 2). It uses the fact that subgroups of the 3-dimensional Heisenberg group are of bounded minimal generation (Theorem 7.4). So T^{2} is the only nilmanifold that does not admit a metric of Hedlund type.

To prove theorem 5.1 we will use:

THEOREM 5.2 (K.A. Hirsch, [6] theorem 2.1). Let Π_{1} be a finitely generated nilpotent group. Then Π_{1} can be embedded as a subgroup into the direct product D = Γ×F where Γ is a torsionfree nilpotent finitely generated group and F is a finite nilpotent group.

The elements in Π_{1} of finite order form a normal subgroup T = F ∩ Π_{1}. Looking at Baumslag’s proof of the theorem of Hirsch, we immediately see that Γ can be chosen as Π_{1}/T and that

\begin{aligned}Π_{1} & Γ × F \\ & Γ = Π_{1}/T \end{aligned}

commutes.

Proof of Theorem 5.1. Suppose Π_{1} to be embedded in D = Γ × F as above.

Now embed Γ as a lattice in the nilpotent Lie-group G (Theorem 4.1).

Abelianisation of the above diagram and tensoring with ℝ shows that

rank \frac{Γ}{[Γ, Γ]} = rank \frac{Π_{1}}{[Π_{1}, Π_{1}]}( = b_{1}),

so there is a natural isomorphism

\frac{G}{[G, G]} ˜→ ℝ^{b_{1}}

that maps the image of Γ to ℤ^{b_{1}}.

It is well-known that for n ≥ 4 any finitely presented group is the fundamental group of an n-dimensional compact manifold ([22] p. 114). So let S be a 4-dimensional compact Riemannian manifold with π_{1}(S) = F.

Now take an ϵ-Hedlund metric on T^{b_{1}} × S with hyperbolic c_{i} and choose a metric on (Γ\G) × S such that

(Γ\G) × S → T^{b_{1}} × S

is a Riemannian submersion and such that [G, G] acts isometrically on G × S. As (Γ\G) × S has fundamental group D, we can find a Riemannian covering M with π_{1}(M) = Π_{1}. So (1), (2) and (3) are fulfilled.

Using Theorem 2.1, Proposition 3.5, Supplement 3.6 and Proposition 2.4 we also get (4). □

In the remaining part of this section we will discuss some related topics and some additional properties of the above examples.

For arbitrary compact manifolds M with Π_{1} := π_{1}(M) we get via the Hurewicz map

H_{1}(M, ℤ) = \frac{Π_{1}}{[Π_{1}, Π_{1}]} and H_{1}(M, ℝ) = H_{1}(M, ℤ) ⊗_{ℤ} ℝ.

Dividing H_{1}(M, ℝ) by the image of H_{1}(M, ℤ) we get a torus T^{b_{1}} in analogy to the above constructions. This torus is known as the Jacobi variety ([16]4.21). H_{1}(M, ℝ) also carries a norm, the “stable norm ∥ . ∥”, induced by the Riemannian structure of M ([12],[16]4.18,[5]). (M̃, ϵ⟨ . , . ⟩) converges in the Gromov-Hausdorff-sense to (H_{1}(M, ℝ), ∥ . ∥) for ϵ → 0 if Π_{1} is abelian. For nilpotent Π_{1} it converges to a Carnot-Caratheodory space ([24]) and the stable norm is essential for measuring distances on this space.

Bangert used the stable norm to prove an existence theorem for minimal geodesics ([4],[5]). As a corollary he proved the existence of at least b_{1} different geodesics such that the “rotation set” R(c_{i}) of each c_{i} contains only one vector in H_{1}(M, ℝ) and ⋃_{i} R(c_{i}) is a basis of H_{1}(M, ℝ). In the above examples the geodesics c_{1}, …, c_{b_{1}} have the properties of the geodesics whose existence has been shown by Bangert: each R(c_{i}) contains only one vector and their union is a basis. For our examples the stable norm written in this basis is just

∥x∥ = ∑_{i=1}^{b_{1}} ϵ_{i}|x^{i}|.  (3)

Equation (3) can be seen from Bangert’s theorem and the characterisation of the minimal geodesics or just using the fact that if a Riemannian submersion p: M_{1} → M_{2} of compact manifolds M_{i} induces an isomorphism p_{#}: H_{1}(M_{1}, ℝ) → H_{1}(M_{2}, ℝ), then p_{#} preserves the stable norm.

This last fact can also be used to construct metrics on nilmanifolds with non-left-invariant metric on the universal covering that have a smooth unit ball of the stable norm. (Just take a suitable lift of a flat metric on the Jacobi variety.)

6. Expressway metrics

In the previous section we proved that for every finitely generated nilpotent group Π_{1} there is a Riemannian manifold M with π_{1}(M) = Π_{1} and only few directions of minimal geodesics. On the other hand, many properties concerning minimal geodesics only depend on the fundamental group and an induced distance on it. Therefore it seems likely that we could find a suitable Riemannian metric on every compact manifold with nilpotent fundamental group. Moreover the metrics in the last section admit continuous families of minimal geodesics if dim[G, G] > 0, whereas Hedlund’s original examples only admit very few ones. So it would be interesting to generalize Hedlund’s methods directly.

In this section we try to use Hedlund’s method (generalized by Bangert [4]) directly to construct Hedlund type metrics on arbitrary (compact) manifolds with nilpotent fundamental group Π_{1}. It turns out that we succeed only if Π_{1} has an algebraic property that we call “bounded minimal generation”. We can prove that lattices in Heisenberg groups have this property but we do not know if this is true for all finitely generated nilpotent groups or not.

The following construction seems to be very special, but the author thinks that it will be difficult to find a different construction without getting a problem similiar to the bounded-minimal-generation-problem described in the next section. We will omit an explicit definition of the metrics and proofs of the statements concerning these metrics as the exact formulae only give little insight in what happens. (For details see [1]). Instead we will give an informal description.

[figure: c_{1}; c_{2}; p]

Figure 3. An expressway with two generators

When the author tried to generalise Bangert’s construction of Hedlund metrics ([4]) to manifolds with arbitrary fundamental groups, he took closed curves c_{1}, …, c_{k} based in p ∈ M, whose homotopy classes [c_{i}] generate Π_{1}. By a small perturbation it is possible to transform the c_{i} into smooth disjoint embeddings of S^{1} ↪ M passing near p if dim M ≥ 3. Now he chose a Riemannian metric that is very small in a small neighborhood of the c_{i} and small in the neighborhood of certain paths joining the c_{i} to p, but relatively big outside these neighborhoods. We can assume that the c_{i} are hyperbolic minimal geodesics of length ϵ.

Roughly speaking, the Riemannian distance looks like the distance, a car driver has in his mind: there are some “expressways” (the neighborhoods of the c_{i} and the joining paths) where normed curves run very fast and in other regions where they move relatively slow. So minimal geodesics run most of their time on these “expressways”. To be more precise the author showed that if c is a minimal geodesic and E the expressway, then the length of each connected component of c^{−1}(M \E) is small. But we do not know whether the total length of c^{−1}(M \ E) is bounded.

So for arbitrary nilpotent fundamental groups the author was unable to get analogues of propositions 3.2 and 3.4. The situation is much nicer when Π_{1} is of bounded minimal generation with respect to the [c_{i}] (Definition 7.1). Here we get analogues of propositions 3.2 and 3.4. The classification of minimal geodesics then can be reduced to combinatorial group properties of Π_{1}, and every minimal geodesic is asymptotic in each of its senses to one of the c_{i}. So the results are similar as on T^{n} (n ≥ 3). Unfortunately the bounded-minimal-generation-problem seems to be hard.

7. Groups of Bounded Minimal Generation

Let S_{Γ} be a finite set of generators of the group Γ, i.e. every γ ∈ Γ can be written as a word (s_{1}, s_{2}, …, s_{l}) with s_{i} ∈ S_{Γ} ˙∪S_{Γ}^{−1}. The number l =: l((s_{1}, s_{2}, …)) is the length of the word, furthermore we define the change number

C((s_{1}, s_{2}, …, s_{l})) := #{i ∈ {1, 2, …, l − 1} | s_{i} ≠ s_{i+1}} + 1
= sup {k | s_{i_{1}} ≠ … ≠ s_{i_{k}}, 1 ≤ i_{1} < … < i_{k} ≤ l} .

This is a group theoretical analogue of Definition 3.3. For the empty word representing the neutral element we set l(∅) = C(∅) = 0.

The word (s_{1}, s_{2}, …) is of minimal length if every (s′_{1}, s′_{2}, …) representing the same γ ∈ Γ satisfies:

l((s_{1}, s_{2}, …)) ≤ l((s′_{1}, s′_{2}, …)).

DEFINITION 7.1. (Γ, S_{Γ}) is a Group of Bounded Minimal Generation (BMG group) if there is a B ∈ ℕ such that every γ ∈ Γ can be represented by a word of minimal length (s_{1}, s_{2}, …) in S_{Γ} ˙∪S_{Γ}^{−1} with C((s_{1}, s_{2}, …)) ≤ B. The minimal such B will be called the bound.

Every finitely generated abelian group together with an arbitrary finite set of generators S_{Γ} is a BMG group with B ≤ #S_{Γ}.

Gromov proved in [15] that every finitely generated group of polynomial growth is virtually nilpotent, i.e. it contains a nilpotent subgroup of finite index. If (Γ, S_{Γ}) is a BMG group, then Γ is of polynomial growth and therefore virtually nilpotent. Yet, it is not clear to the author if the converse holds.

OPEN PROBLEM 7.2. Is every finitely generated virtually nilpotent group a BMG group?

Remark. It is not even clear whether the BMG property is independent of the choice of the set of generators.

Constructing new BMG groups from old ones. It is staightforward to show:

1. If (Γ, S_{Γ}) and (Γ′, S_{Γ′}) are BMG groups with bounds B and B′, then (Γ × Γ′, S_{Γ} ∪ S_{Γ′}) is a BMG group with bound ≤ B + B′.

2. If (Γ, S_{Γ}) is a BMG-group with bound B and h: Γ → Γ′ a group homomorphism, then (h(Γ), h(S_{Γ})) is a BMG-group with bound ≤ B.

3. If Γ_{2} is the semidirect product of a BMG group (Γ_{1}, S_{Γ_{1}}) and a finite group, then there is a generating system S_{Γ_{2}} of Γ_{2} such that (Γ_{2}, S_{Γ_{2}}) is a BMG group.

Heisenberg groups. For m ∈ ℕ, p⃗, q⃗ ∈ ℝ^{m}, z ∈ ℝ we define the matrix

M(p⃗, q⃗, z) := ( \begin{matrix}1 & p^{t} & z \\ 0 & 1_{m} & q⃗ \\ 0 & 0 & 1 \end{matrix} )

H_{m} := {M(p⃗, q⃗, z) | p⃗, q⃗ ∈ ℝ^{m}, z ∈ ℝ} is the 2m + 1-dimensional Heisenberg group.

For r = (r_{1}, …, r_{m}) ∈ ℕ^{m} such that r_{i} divides r_{i+1}, 1 ≤ i < m, we set

rℤ^{m} := {(r_{1}x_{1}, …, r_{m}x_{m}) | x_{j} ∈ ℤ}
Γ_{r} := {M(p⃗, q⃗, z) | p⃗ ∈ rℤ^{m}, q⃗ ∈ ℤ^{m}, z ∈ ℤ} .

These Γ_{r} are lattices in H_{m}, i.e. discrete, cocompact subgroups.

THEOREM 7.3 ([13], §2.). For every lattice Γ of H_{m} there exists a unique r and an automorphism of H_{m} mapping Γ to Γ_{r}.

THEOREM 7.4. Every lattice Γ of H_{m} has a set of generators S, such that (Γ, S) is a BMG-group.

Remark. It is not difficult to show that any discrete subgroup of H_{m} is of the form Γ′ × ℤ^{k}, where Γ′ is trivial or a lattice in a Heisenberg group. So Theorem 7.4 immediately generalizes to discrete subgroups.

LEMMA 7.5. Γ_{1} = {M(p, q, z) | p, q, z ∈ ℤ} ⊂ H_{1} together with S_{Γ_{1}} := {M(1, 0, 0), M(0, 1, 0)} is a BMG-group.

Proof of Theorem 7.4. We can assume Γ = Γ_{r}. We denote the standard basis of ℝ^{m} by e_{1}, …, e_{m}. The mappings

f_{i}: H_{1} → H_{m}
M(p, q, z) ↦ M(pr_{i}e_{i}, qe_{i}, zr_{i})

for i = 1, …, m and

f_{0}: ℝ → H_{m}
z ↦ M(0, 0, z)

define a group epimorphism

f: ℝ × X_{i=1}^{m} H_{1} → H_{m}
(z, h_{1}, …, h_{m}) ↦ f_{0}(z)f_{1}(h_{1})f_{2}(h_{2}) … f_{m}(h_{m})

that maps ℤ × X_{i=1}^{m} Γ_{1} to Γ_{r}. Using lemma 7.5 we know that (Γ_{1}, S_{Γ_{1}}) is a BMG-group. Applying the above constructions of new BMG-groups we see that (Γ, S) is a BMG-group for S := f(⋃_{i=0}^{m} S_{i}), where S_{0} is 1 ∈ ℝ and S_{i} is S_{Γ_{1}} in the i-th slot. □

Proof of Lemma 7.5. We set g_{1} := M(1, 0, 0), g_{2} := M(0, 1, 0), then [g_{1}, g_{2}] = g_{1}g_{2}g_{1}^{−1}g_{2}^{−1} = M(0, 0, 1). For every word w in the generators g_{1} and g_{2} we will explain how to construct a word w′ of minimal length and with C(w′) ≤ 6 representing the same γ ∈ Γ.

In order to construct w′ we will give a geometric interpretation of the problem. To every word w we will associate inductively a path p(w) in

P := \{( \begin{matrix}x \\ y \end{matrix} ) ∈ ℝ^{2} | x ∈ ℤ or y ∈ ℤ\} .

At first we associate to w = ∅ the constant path in ( \begin{matrix}0 \\ 0 \end{matrix} ), and to

w = g_{1} (resp. w = g_{2}, g_{1}^{−1}, g_{2}^{−1}) the straight line from ( \begin{matrix}0 \\ 0 \end{matrix} ) to ( \begin{matrix}1 \\ 0 \end{matrix} )

(resp. ( \begin{matrix}0 \\ 1 \end{matrix} ) , ( \begin{matrix}−1 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ −1 \end{matrix} )). Then we associate to the word w_{1}w_{2} the

path p(w_{1}w_{2}) that consists of the path p(w_{1}) and then p(w_{2}), translated

by the endpoint of p(w_{1}) — we get a path from ( \begin{matrix}0 \\ 0 \end{matrix} ) to the sum of the

endpoints of p(w_{1}) and p(w_{2}).

Now we define

I(w) = ∫_{p(w)} x dy.

Let (i_{1}, i_{2}) be the endpoint of p_{w}. Then w represents g_{2}^{i_{2}}g_{1}^{i_{1}} [g_{1}, g_{2}]^{I(w)}, l(w) = L(p(w)), the change number C(w) of w is equal to the number of direction changes of p(w) plus 1.

We get an geometric interpretation of I(w) by applying Stoke’s theorem:

I(w) = ∫_{p(w)} x dy = ∫_{p(wg_{1}^{−i_{1}}g_{2}^{−i_{2}})} x dy = ∫_{A} dx ∧ dy,

where A is the 2-chain whose boundary is p (wg_{1}^{−i_{1}}g_{2}^{−i_{2}}). As dx ∧ dy is the volume element of ℝ^{2}, we interpret I(w) as the oriented area of A.

We get an isoperimetric problem on P: Finding words of minimal length means finding paths of minimal length in P with given integral ∫ x dy.

With this geometric interpretation the lemma is almost obvious. Using symmetries we can assume i_{1}, i_{2} ≥ 0, I(w) ≥ 0. Consider the special cases

1. I(w) ≤ i_{1} · i_{2},

2. i_{1}i_{2} < I(w) ≤ max {i_{1}, i_{2}}^{2},

3. max {i_{1}, i_{2}}^{2} < I(w).

In each case we can find a word w′ of minimal length, equivalent to w with C(w′) ≤ 6. □

Remark. Michael Stoll [28] treats the continuous analogue of bounded minimal generation for nilpotent Lie-groups. He proves that every 2-step nilpotent Lie-group fulfills it, but he states that there are counterexamples for 3-step nilpotent Lie-groups.

Remark. Several authors ([9],[29],[26] and [27]) consider definitions (“groups of bounded generation”, “groups of finite width”) which are formally related to our groups of bounded minimal generation. The main difference is that these notions use arbitrary not only minimal representatives.

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