ON DENSE FREE SUBGROUPS OF LIE GROUPS

E. BREUILLARD, T. GELANDER

Abstract. We investigate dense subgroups in real Lie groups. In particular we show that any dense subgroup of a connected semisimple real Lie group G contains a free group on two generators which is still dense in G, and that any finitely generated dense subgroup in a connected nonsolvable Lie group H contains a dense free subgroup of rank ≤ 2 · dim H.

1. Introduction

The main purpose of this paper is to give a proof of the following statement, which answers a question raised by A. Lubotzky and R. Zimmer.

Theorem 1.1. Let G be a connected semi-simple real Lie group and Γ a dense subgroup of G. Then Γ contains two elements which generate a dense free subgroup of G.

We shall also give the following generalizations of this result

Theorem 1.2. Let G be a connected real Lie group which is topologically perfect (i.e. the commutator [G, G] is dense in G). If its Lie algebra 𝔤 is generated by l elements, then any dense subgroup contains a dense free sub-subgroup of rank l.

Theorem 1.3. Let G be a connected non solvable real Lie group of dimension d. Then any finitely generated dense subgroup of G contains a dense free subsubgroup of rank 2d.

The finite generation assumption in 1.3 is crucial. Moreover, for some particular groups G one can give a smaller bound for the rank of the dense free sub-subgroup than the bounds l and 2d given in 1.2 and 1.3. In fact, as it will be clear from the proofs, one can find a free dense sub-subgroup of rank k, for any integer k ≥ l in theorem 1.2 (resp. k ≥ 2d in theorem 1.3). We will give examples to illustrate these facts.

A celebrated theorem of Tits (see [12], Theorem 1) asserts that any Zariskidense subgroup of a semisimple algebraic group over a field of characteristic zero contains a free subgroup on two generators. The difficulty of the problem we are considering comes from the fact that the free group obtained in Tits’ proof is in general discrete in the Lie group G. The proof of Theorem 1.1 relies on a careful study of the so-called ”proximal” elements in Γ acting on a projective space over some local field. Unlike in Tits’ paper, where powers of suitable semisimple elements are used to produce proximal elements, our method is inspired from that of Abels-Margulis-Soifer (see [1]), which is based on the Cartan decomposition of projective transformations.

Date: January 2002.

In section 2, we give a simple method for producing finitely generated dense subgroups in connected real Lie groups. Section 3 is devoted to the study of proximal transformations in projective spaces over local fields. We give some quantitative estimates that enable us to exhibit proximal elements with nice properties. In section 4 we explain how to generate elements of the subgroup Γ with a nice proximal action on some projective space. Proposition 4.1 gives a practical way of generating very proximal elements in the group Γ. In particular, one can easily derive from it the original statement of Tits’ alternative. Finally, the last section contains the proof of the three theorems above.

Let us make one remark about terminology.

Remark 1.1. In this paper, we use both the usual real topology (or the Hausdorff topology induced by a local field k) and the Zariski topology on the groups considered. To avoid confusion, we shall add the prefix Zariski- to any topological notion regarding the Zariski topology (Zariski-connected, Zariskidense, etc.). For the real topology, however, we shall plainly say ”dense” or ”open” without further notice. Note that the Zariski topology on rational points does not depend on the field of definition, i.e. if V is an algebraic variety defined over a field K and if L is any extension of K, then the K-Zariski topology on V (K) coincide with the trace of the L-Zariski topology on it.

2. Generating dense subgroups in Lie groups

We call a connected Lie group G topologically perfect if its commutator group [G, G] is dense, or equivalently, if G has no continuous surjective homomorphism to the circle.

The following result shows that, in a topologically perfect group, elements which lay near the identity generate a dense subgroup, unless they have some algebraic reason not to. Similar statements were established by Kuranishi (see [8]).

Theorem 2.1 (Generating dense subgroups in topologically perfect groups). Let G be a connected topologically perfect real Lie group with Lie algebra 𝔤. Then there is an identity neighborhood Ω ⊂ G, on which log = exp^{−1} is a well defined diffeomorphism, such that g_{1}, …, g_{m} ∈ Ω generate a dense subgroup whenever log(g_{1}), …, log(g_{m}) generate 𝔤.

Proof. Recall that a Zassenhaus neighborhood of a real Lie group is an identity neighborhood Ω for which the intersection Ω ∩ Σ with any discrete subgroup Σ is contained in a connected nilpotent Lie subgroup. A classical theorem of Kazhdan and Margulis (see [11] 8.16) says that a Zassenhaus neighborhood always exists. Checking the details of the proof of the Kazhdan-Margulis theorem, one can easily verify that the identity neighborhood Ω established there, satisfies the stronger property that its image under any surjective homomorphism f is a Zassenhaus neighborhood in the image group (this is because the properties of the neighborhoods established in 8.18 and 8.20 in [11] are inherited to their images). Moreover, as in [11], one writes Ω = exp V for a suitable neighborhood V of 0 in 𝔤, and then, a subset {X_{1}, …, X_{p}} ⊂ (df)(V) generates a nilpotent Lie subalgebra iff exp(X_{1}), …, exp(X_{p}) generate a nilpotent subgroup in f(G). We shall call such an Ω a strongly Zassenhaus neighborhood.

Let Ω be a strongly Zassenhaus neighborhood in G, and assume that it is small enough so that log |_{Ω} is a well defined diffeomorphism. Pick g_{1}, …, g_{m} ∈ Ω for which log(g_{1}), …, log(g_{m}) generate 𝔤. Denote by A the closure of ⟨g_{1}, …, g_{m}⟩, and by A^{0} its identity component. We need to show that A^{0} = G.

First observe that A^{0} is normal in G, i.e. its Lie subalgebra 𝔞 is an ideal in 𝔤. Indeed, since 𝔤 = ⟨log(g_{i})⟩, it is enough to show that ad( log(g_{i}))(𝔞) = 𝔞 for any 1 ≤ i ≤ m, but

ad(log g_{i})(𝔞) = ( log Ad(g_{i}))(𝔞),

and since g_{i} ∈ A, it normalizes A^{0}, i.e. Ad(g_{i})(𝔞) = 𝔞, which implies also that ( log Ad(g_{i}))(𝔞) = 𝔞.

Next consider the quotient map f : G → G/A^{0}. Clearly, f(⟨g_{1}, …, g_{m}⟩) = f(A) is discrete, and generated by the set {f(g_{1}), …, f(g_{m})} which is contained in the Zassenhaus neighborhood f(Ω). Therefore f(⟨g_{1}, …, g_{m}⟩) is a nilpotent group, but this means that

log f(g_{i}) = (df)(log g_{i}), i = 1, …, m

generate a nilpotent Lie algebra. As {(df)(log g_{i})}_{i=1}^{m} generate 𝔤/𝔞, this implies that G/A^{0} is nilpotent. Since G is assumed to be topologically perfect, any nilpotent quotient is trivial. Thus G = A^{0} and ⟨g_{1}, …, g_{m}⟩ is dense. □

In fact, it is enough to require that log(g_{1}), …, log(g_{m}) generate a Lie subalgebra which corresponds to a dense Lie subgroup. This phenomenon is illustrated by the following

Example 2.1. Let S̃L_{2}(ℝ) be the universal covering of SL_{2}(ℝ), and let a ∈ S̃L_{2}(ℝ) be a central element of infinite order. Let α be an irrational rotation in the circle group 𝕋. Consider the group

G = (S̃L_{2}(ℝ) × 𝕋)/⟨(a, α)⟩.

G is an example of a topologically perfect non-perfect group. Its Lie algebra 𝔤 ≅ sl_{2}(ℝ) ⊕ ℝ is not generated by 2 elements, but the 2-generated Lie subalgebra sl_{2}(ℝ) corresponds to a dense Lie subgroup in G. Theorem 1.2 guarantees only that any dense subgroup of G contains a dense F_{3}, however, it is possible to show (by the same argument which proves Theorem 1.1) that there is always a dense F_{2}.

However, in general, 2 generators are not enough. This phenomenon is illustrated by the following

Example 2.2. Consider the complex Lie group G = SL_{2}(ℂ) · (ℂ^{2})^{5}, with the ordinary action on SL_{2}(ℂ) on each of the 5 copies of ℂ^{2}. It is easy to verify that

(1) G is perfect, i.e. [G, G] = G.

(2) The exponential map is onto.

(3) Any 2 elements in G’s Lie algebra 𝔤 = sl_{2}(ℂ) · (ℂ^{2})^{5} generate a Lie subalgebra for which the closure of the corresponding Lie subgroup is proper.

These properties imply that there is no 2-generated dense subgroup in G.

It is well known that a real semisimple Lie algebra is generated by 2 elements (cf. [8] or [3] VIII, 3, ex. 10). We therefore obtain

Theorem 2.2 (Generating dense subgroups in semisimple groups). Let G be a connected semisimple Lie group. Then there exists an identity neighborhood Ω ⊂ G, and a proper exponential algebraic subvariety R ⊂ Ω × Ω such that ⟨x, y⟩ is dense in G whenever (x, y) ∈ Ω × Ω \ R.

Theorem 2.1 yields a lot of freedom in the procedure of generating dense subgroups in topologically perfect groups. In fact, if G is a topologically perfect Lie group and g_{1}, …, g_{m} ∈ G are elements close enough to 1 for which log(g_{1}), …, log(g_{m}) generate the Lie algebra 𝔤, and if U ⊂ G is a small enough identity neighborhood, then for any selections h_{i} ∈ g_{i} · U, log(h_{1}), …, log(h_{m}) generate 𝔤, and h_{1}, …, h_{m} generate a dense subgroup in G. In particular, we obtain Corollary 2.3. If G is a topologically perfect Lie group for which the Lie algebra is generated by m elements, then any dense subgroup in G contains a m-generated dense subgroup.

Proof. Let g_{1}, …, g_{m} and U be as in the last paragraph above. Clearly, if Γ ≤ G is a dense subgroup then Γ∩g_{i}·U ≠ ∅. Pick h_{i} ∈ Γ∩g_{i}·U, i = 1, …, m then ⟨h_{1}, …, h_{m}⟩ is dense. □

The requirement that G is topologically perfect is crucial for corollary 2.3 as explained by the following remark:

Remark 2.4. If a connected Lie group G is not topologically perfect then it has a surjective homomorphism to the circle. Let Γ ≤ G be the pre-image of the group of rational rotations. Clearly Γ is dense in G but, as any finitely generated group of rational rotations is finite, Γ has no finitely generated dense subgroup.

This remark explains why, when G is not topologically perfect, we shall consider only finitely generated dense subgroups. Then we can control the number of generators, by the following reasoning.

Proposition 2.5. Let G be a connected Lie group and Γ ≤ G a finitely generated dense subgroup. Then Γ contains a dense subgroup on 2 dim G generators. If G is compact then Γ contains a dense subgroup on dim G generators.

Proof. Assume first that G is abelian. Then G = ℝ^{d_{1}} × 𝕋^{d_{2}} and we can find d_{1} elements in Γ which generate a discrete cocompact subgroup. Dividing by this subgroup, we may assume that G is a torus. Then we argue by induction on dim G. As Γ is finitely generated and dense it contains an element γ of infinite order. By replacing γ by some power γ^{j} if necessary, we obtain an element which generates a subgroup with connected closure C of positive dimension. By induction, the proposition holds for G/C. Lift arbitrarily to G a set of dim G/C generators for a dense subgroup of the image of Γ in G/C. Together with γ they generate a dense subgroup in G.

For the general case, we argue as follows. Consider the sequence G_{i+1} = \overline{[G_{i}, G_{i}]}, where [G_{i}, G_{i}] is the commutator group of G_{i}, starting at G_{0} = G. As dim G is finite this sequence stabilizes at some finite step k. Then G_{k} is a connected closed normal topologically perfect subgroup. Moreover the commutator [Γ, Γ] is a finitely generated dense subgroup in G_{1}, and similarly, the i’th commutator of Γ is a finitely generated dense subgroup in G_{i}. Thus, by the above paragraph, we can find, for each i, 0 ≤ i < k, a set Σ_{i} of 2 dim G_{i}/G_{i+1} (or dim G_{i}/G_{i+1} in the compact case) elements in G_{i}∩Γ which projects to generators of a dense subgroup in the abelian group G_{i}/G_{i+1}. Additionally, by corollary 2.3 there is a set Σ_{k} of dim G_{k} elements in Γ ∩ G_{k} which generate a dense subgroup of G_{k} (in the non-abelian compact case G_{k} = G_{1} is semisimple and we can take Σ_{k} of size 2).

Clearly ⋃_{i=0}^{k} Σ_{i} generates a dense subgroup and has the required cardinality. □

3. Projective transformations, proximality, ping-pong

Let k be a local field equipped with an absolute value | · |. We wish to investigate the action of projective transformations in PSL_{n}(k) on the projective space ℙ^{n−1}(k). We shall start by introducing a nice metric on the projective space ℙ^{n−1}(k), and then consider the Cartan decomposition, i.e. the KAK decomposition in SL_{n}(k).

(1) Consider first the case when k is arcimedean, i.e. k = ℝ or ℂ. We shall denote by ∥·∥ the canonical euclidean (resp. hermitian) norm on k^{n}, i.e.∥x∥^{2} = ∑ |x_{i}|^{2} if x = ∑ x_{i}e_{i}, where (e_{1}, ..., e_{n}) is the canonical basis of k^{n}. The Cartan decomposition in SL_{n}(k) reads

SL_{n}(k) = KAK

where,

K = SO_{n}(ℝ) or SU_{n}(ℂ) acording to whether k = ℝ or ℂ,
and A = {diag(a_{1}, …, a_{n}) : a_{1} ≥ … ≥ a_{n} > 0, ∏ a_{i} = 1}.

Any element g ∈ 𝕊𝕃_{n}(ℝ) can be decomposed as a product g = k_{g}a_{g}k′_{g}, where k_{g}, k′_{g} ∈ K and a_{g} ∈ A. We remark that a_{g} is uniquely determined by g, but k_{g}, k′_{g} are not. Even so, we shall use the subscript g to indicate the relation to g.

(2) Next consider the case where k is non-archimedean with valuation ring O_{k} and uniformizer π. We endow k^{n} with the canonical norm defined by ∥x∥ = max |x_{i}| , where x = ∑ x_{i}e_{i} is the expansion of x with respect to the canonical basis (e_{1}, ..., e_{n}) of k^{n}. Then we get

SL_{n}(k) = KAK

where

K = SL_{n}(O_{k}),
and A = {diag(π^{j_{1}}, …, π^{j_{n}}) : j_{i} ∈ ℤ, j_{i} ≥ j_{i+1}, ∑ j_{i} = 0}.

Any g ∈SL_{n}(k) can be decomposed as a product g = k_{g}a_{g}k′_{g}, where k_{g}, k′_{g} ∈ K and a_{g} ∈ A (see [10] page 150, or [4] 4.4.3.). Again a_{g} is uniquely determined, but k_{g}, k′_{g} are not.

In both cases, the canonical norm on k^{n} gives rise to the associated canonical norm on ⋀^{2} k^{n}. We shall then define the standard metric on ℙ^{n−1}(k) by the formula

d([v], [w]) = \frac{∥v ∧ w∥}{∥v∥ · ∥w∥}

This is well defined and satisfies the following properties:

• d is a distance on ℙ^{n−1}(k) which induces the canonical topology inherited from the local field k.

• d is an ultra-metric distance if k is non-archimedean, i.e.

d([v], [w]) ≤ max{d([v], [u]), d([u], [w])}

for any non-zero vectors u, v and w in k^{n}.

• If f is a linear form k^{n} → k, then for any non-zero vector v ∈ k^{n},

(1)  d([v], [ker f]) = \frac{|f(v)|}{∥f∥ · ∥v∥}

• the compact group K acts by isometries on (ℙ^{n−1}(k), d). In the sequel we shall denote by a_{1}(g), ..., a_{n}(g) the coefficients of the

diagonal matrix a_{g} corresponding to g in the Cartan decomposition. For further use, we note that, in the above notations for any matrix g ∈SL_{n}(k),

∥g∥ = a_{1}(g)
∥⋀^{2} g∥ = a_{1}(g)a_{2}(g).

Let us observe a few properties of projective transformations. For g ∈ SL_{n}(k) we denote by [g] the corresponding projective transformation [g] ∈ PSL_{n}(k).

Lemma 3.1. Every projective transformation is bi-Lipschitz on the entire projective space for some constant depending on the transformation.

Proof. As the compact group K acts by isometries on ℙ^{n−1}(k), the KAK decomposition allows us to assume that g = a_{g} = diag(a_{1}, ..., a_{n}) ∈ A. Thus,

one only needs to check the easy fact that a_{g} is |\frac{a_{1}}{a_{n}} |^{2}-Lipschitz. □

Definition 3.2. Let ϵ ∈ (0, 1). A projective transformation [g] ∈ PSL_{n}(k) is called ϵ-contracting if there exist a point v_{g} ∈ ℙ^{n−1}(k), called an attracting point of [g], and a projective hyperplane H_{g}, called a repulsive hyperplane of [g], such that [g] maps the complement of the ϵ-neighborhood of H_{g} ⊂ ℙ^{n−1}(k) into the ϵ-ball around v_{g}. We say that [g] is ϵ-very contracting if both [g] and [g^{−1}] are ϵ-contracting.

Note that in general v_{g} and H_{g} are not necessarily unique. Nevertheless, all the statements we shall make about them will be valid for any choice.

The following proposition shows that “ϵ-contraction” is equivalent to ”a big ratio between the first and second diagonal terms in the KAK decomposition.”

Proposition 3.1. Let ϵ < \frac{1}{4}. If |\frac{a_{2}(g)}{a_{1}(g)}| ≤ ϵ^{2}, then [g] is ϵ-contracting. More precisely, writing g = k_{g}a_{g}k′_{g}, one can take H_{g} to be the projective hyperplane spanned by {k′_{g}^{−1}(e_{i})}_{i=2}^{n}, and v_{g} = [k_{g}(e_{1})].

Conversely, suppose [g] is ϵ-contracting and k is non-archimedean with uniformizer π (resp. archimedean), then |\frac{a_{2}(g)}{a_{1}(g)}| ≤ \frac{ϵ^{2}}{|π|} (resp. |\frac{a_{2}(g)}{a_{1}(g)}| ≤ 4ϵ^{2}).

Proof. Suppose that |\frac{a_{2}(g)}{a_{1}(g)}| ≤ ϵ^{2}. Using the Cartan decomposition and the fact that K acts by isometries, we can again assume that

g = a_{g} = diag(a_{1}, ..., a_{n})

Let [v] ∈ ℙ^{n−1}(k) be outside the ϵ-neighborhood of H_{g}, which, in our case, simply means that (cf. 1)

d([v], H_{g}) = \frac{|v_{1}|}{∥v∥} ≥ ϵ

Then

d([gv], [e_{1}]) = \frac{∥gv ∧ e_{1}∥}{∥gv∥} ≤ \frac{|a_{2}| · ∥v∥}{|a_{1}| · ∥v_{1}∥} ≤ ϵ

since ∥gv ∧ e_{1}∥ ≤ |a_{2}| ∥v∥ and ∥gv∥ ≥ |a_{1}v_{1}|.

The converse is more delicate. We shall describe the non-archimedean case, and remark that the archimedean case can be dealt in an analogous way. Again we can take g = a_{g}. Let f = (f_{1}, ..., f_{n}) be a linear form of norm 1 such that ker(f) = H_{g} and let w ∈ k^{n} be a normalized representative of the attracting point of v_{g}. The fact that [g] is ϵ-contracting means that for every non-zero vector v ∈ k^{n}

(2)  |f(v)| ≥ ϵ ∥v∥ ⇒ ∥gv ∧ w∥ ≤ ϵ ∥gv∥ .

Suppose that for some index i_{0}, |f_{i_{0}}| ≥ ϵ. Take v = e_{i_{0}} and get that

|a_{i_{0}}| ∥e_{i_{0}} ∧ w∥ ≤ ϵ |a_{i_{0}}|

So ∥e_{i_{0}} ∧ w∥ = max_{i≠i_{0}} |w_{i}| ≤ ϵ and hence |w_{i_{0}}| = 1, and i_{0} is unique. Since ∥f∥ = max |f_{i}| = 1, we must then have |f_{i_{o}}| = 1 and max_{i≠i_{0}} |f_{i}| < ϵ.

We now claim that i_{0} = 1. Suppose the contrary and take v = e_{1} + e_{i_{0}}. Then ∥v∥ = 1 and f(v) = f_{1} + f_{i_{0}}. Since |f_{i_{0}}| = 1 ≥ ϵ > |f_{1}| we indeed have

|f(v)| ≥ ϵ ∥v∥

Hence, by (2), we obtain

∥a_{1}e_{1} ∧ w + a_{i_{0}}e_{i_{0}} ∧ w∥ ≤ ϵ ∥a_{1}e_{1} + a_{i_{0}}e_{i_{0}}∥

which also reads

|a_{1}| ≤ ϵ |a_{1}|

and gives the desired contradiction.

Finally take v = xe_{1} + e_{2}, where x ∈ k is chosen such that |x| is least possible and ≥ ϵ. Then ∥v∥ = 1 and f(v) = xf_{1} + f_{2}. Since |xf_{1}| ≥ ϵ > |f_{2}| we have

|f(v)| ≥ ϵ ∥v∥

and, again, by (2), we obtain

∥xa_{1}e_{1} ∧ w + a_{2}e_{2} ∧ w∥ ≤ ϵ ∥xa_{1}e_{1} + a_{2}e_{2}∥

Suppose |a_{2}| > ϵ |x| |a_{1}| then the last inequality translates to

|a_{2}| ≤ ϵ max{|xa_{1}| , |a_{2}|}

which leads to either |a_{2}| ≤ ϵ |a_{2}|, which is absurd, or |a_{2}| ≤ ϵ |x| · |a_{1}|, which contradicts the assumption.

Therefore we have obtained that |a_{2}| ≤ ϵ |x| |a_{1}|, which, considering the choice of x, implies the desired conclusion:

|\frac{a_{2}}{a_{1}} | ≤ \frac{ϵ^{2}}{|π|}

□

Note that the factor |π| is not necessary when ϵ belongs to the value group of k.

Lemma 3.3. Let r, ϵ ∈ (0, 1]. If |\frac{a_{2}(g)}{a_{1}(g)}| ≤ ϵ, then [g] is \frac{ϵ}{r^{2}} -Lipschitz outside the r-neighborhood of the repulsive hyperplane H_{g} = [span{k′_{g}^{−1}(e_{i})}_{i=2}^{n}].

Proof. Again, we can assume g = a_{g}, v_{g} = [e_{1}], H_{g} = [span(e_{i})_{i≥2}]. Let v, w be arbitrary non-zero vectors in k^{n}. Then ∥gv ∧ gw∥ ≤ |a_{1}a_{2}| ∥v ∧ w∥ and ∥gv∥ ≥ |a_{1}v_{1}| and ∥gw∥ ≥ |a_{1}w_{1}|. Therefore

(3)  d([gv], [gw]) ≤ \frac{|a_{2}| · ∥v∥ · ∥w∥}{|a_{1} · |v_{1}| · |w_{1}|}d([v], [w]).

We conclude by observing that the r-neighborhood of H_{g} = [span(e_{i})_{i≥2}] corresponds to non-zero vectors v for which |v_{1}| ≤ r ∥v∥. □

Note in particular that if δ > 0, any projective transformation [g] ∈PSL_{n}(k) is (1 + δ)-Lipschitz in some open set on ℙ^{n−1}(k).

The following lemma gives a converse statement to the last result as well as a handy criterion for ϵ-contraction.

Lemma 3.4. Let [g] be a projective transformation corresponding to g ∈ SL_{n}(k), and assume that the restriction of [g] to some open set O ⊂ ℙ^{n−1}(k) is ϵ-Lipschitz for some ϵ < 1, then |\frac{a_{2}(g)}{a_{1}(g)}| ≤ ϵ when k is non-archimedean,

and |\frac{a_{2}(g)}{a_{1}(g)}| ≤ \frac{ϵ}{\sqrt{1−ϵ^{2}}} when k is archimedean.

Proof. Once again, we may assume g = a_{g}. Let v be a non-zero vector in k^{n} such that [v] ∈ O. For some δ ∈ k\{0} small enough, both [w_{1}] = [v + δe_{1}] and [w_{2}] = [v + δe_{2}] belong to O. Hence, for w_{1}

(4) |δa_{1}| \frac{∥av ∧ e_{1}∥}{∥av∥ · ∥aw_{1}∥} = d([av], [aw_{1}]) ≤ ϵ · d([v], [w_{1}]) = ϵ |δ| \frac{∥v ∧ e_{1}∥}{∥v∥ · ∥w_{1}∥}

thus

\frac{∥av ∧ e_{1}∥}{∥av∥} ≤ ϵ

In the non-archimedean case, this implies |a_{1}v_{1}| = ∥av∥, and in the archimedean case, ∥av ∧ e_{2}∥ ≥ |a_{1}v_{1}| ≥ \sqrt{1 − ϵ^{2}} ∥av∥. Now, expressing the Lipschitz condition for v and w_{2} as in (4) yields

|a_{2}| \frac{∥av ∧ e_{2}∥}{∥av∥} ≤ ϵ |a_{1}|

which, in the non-archimedean case gives |a_{2}/a_{1}| ≤ ϵ and in the archimedean case, |a_{2}/a_{1}| ≤ ϵ/\sqrt{1 − ϵ^{2}}. □

Definition 3.1. A projective transformation [g] ∈PSL_{n}(k) is called (r, ϵ)- proximal (r > 2ϵ > 0) if it is ϵ-contracting with respect to some attracting point v_{g} ∈ ℙ^{n−1}(k) and some repulsive hyperplane H_{g}, such that d(v_{g}, H_{g}) ≥ r. The transformation [g] is called (r, ϵ)-very proximal if both [g] and [g]^{−1} are (r, ϵ)-proximal.

Similar notions of (r, ϵ)-proximality were defined in [1] and [2].

Definition 3.2. A finite subset F ⊂ PSL_{n}(k) is called (m, r)-separating (r > 0, m ∈ ℕ) if for every choice of 2m points v_{1}, ..., v_{2m} in ℙ^{n−1}(k) and 2m projective hyperplanes H_{1}, ..., H_{2m} there exists γ ∈ F such that

min_{1≤i,j≤2m}{d(γv_{i}, H_{j}), d(γ^{−1}v_{i}, H_{j})} > r.

We use the properties described above in order to construct ϵ-very contracting and (r, ϵ)-very proximal elements from two basic ingredients : a ϵ-contracting element and a r-separating set.

Proposition 3.2. Let F be a (1, r)-separating set (r < 1) in PSL_{n}(k) with uniform bi-Lipschitz constant C = max{biLip([f]) : [f] ∈ F } (see Lemma 3.1).

(i) If ϵ < \frac{r}{2C} and [g] ∈PSL_{n}(k) is an ϵ-contracting transformation, then one can find [f] ∈ F such that [fg] is (r, Cϵ)-proximal.

(ii) Let d = 4 when k is archimedean and d = \frac{1}{|π|} when k is non-archimedean. If ϵ < \frac{r}{\sqrt{2Cd}} and [g] ∈PSL_{n}(k) is ϵ-contracting, then there is an element [f] ∈ F, such that

[gfg^{−1}]

is \frac{\sqrt{2Cd}}{r}ϵ-very contracting.

(iii) If ϵ < \frac{r}{2C^{2}} and [g] ∈PSL_{n}(k) is an ϵ-very contracting transformation, there is an element [f] ∈ F, such that [gf] is (\frac{r}{C} , Cϵ)-very proximal.

Proof. Let v_{g} (resp. H_{g}) be an attracting point (resp. repulsive hyperplane) corresponding to the transformation [g]. Let [f] ∈ F be such that d([f]v_{g}, H_{g}) > r. Then [fg] is clearly Cϵ-contracting with attracting point [f]v_{g} and repulsive hyperplane H_{g}. The choice of f guarantees that [fg] is (r, Cϵ)-proximal. This proves (i).

By Lemma 3.3 and the remark following it, there is an open set O on which [g^{−1}] is, say, \sqrt{2}-Lipschitz. Let [u] be some point inside O. We can find [f] ∈ F such that d([fg^{−1}u], H_{g}) > r and d([f^{−1}g^{−1}u], H_{g}) > r. Since [g] is ϵ-contracting, by Proposition 3.1 we have |\frac{a_{2}(g)}{a_{1}(g)} | ≤ dϵ^{2}. Hence by Lemma 3.3,

[g] is \frac{dϵ^{2}}{r^{2}} -Lipschitz outside the r-neighborhood of H_{g}. Therefore both [gfg^{−1}] and [gf^{−1}g^{−1}] are \frac{dC\sqrt{2}ϵ^{2}}{r^{2}} –Lipschitz in some open neighborhood of [u], which implies by Lemma 3.4, that they both satisfy \frac{|a_{2}|}{|a_{1}|} ≤ 2\frac{dCϵ^{2}}{r^{2}} . By Proposition 3.1

it now follows that they are both \frac{\sqrt{2Cd}}{r}ϵ-contracting transformations. This proves (ii).

Find [f] ∈ F which takes v_{g} at least r away from H_{g} for which [f^{−1}] takes v_{g^{−1}} at least r away from H_{g^{−1}}. Then [gf] is (\frac{r}{C} , Cϵ)-proximal and [f^{−1}g^{−1}] is (r, Cϵ)-proximal. Hence [gf] is (\frac{r}{C} , Cϵ)-very proximal.

□

We end this section by showing how to obtain generators of a free group via the so-called ping-pong lemma, once we are given a very contracting element and a separating set. The following definition and lemma are classical (cf. [12]).

Definition 3.5 (Ping-pong pair). Let k be a local field and V a finite dimensional k-vector space. A pair of projective transformations a, b ∈PSL(V) is called a ping-pong pair if both a and b are (r, ϵ)-very proximal, with respect to some r > 2ϵ > 0, and if the attracting points of a and a^{−1} (resp. of b and b^{−1}) are at least r-apart from the repulsive hyperplanes of b and b^{−1} (resp. of a and a^{−1}).

More generally, a m-tuple of projective transformations a_{1}, …, a_{m} is called a ping-pong m-tuple if all a_{i}’s are (r, ϵ)-very proximal (for some r > 2ϵ >

0) and the attracting points of a_{i} and a_{i}^{−1} are at least r-apart from the repulsive hyperplanes of a_{j} and a_{j}^{−1}, for any i ≠ j.

We have following variant of the ping-pong lemma (see [12] 1.1).

Lemma 3.6. If a_{1}, …, a_{m} ∈PSL(V) forms a ping-pong m-tuple, then ⟨a_{1}, …, a_{m}⟩ is a free group.

Lemma 3.7. Given a_{1}, …, a_{m} ∈ PSL(V), a (m, r)-separating set F, and a ϵ-very contracting element γ (with ϵ < \frac{r}{2c^{4}} , where c is the maximal bi-Lipschitz constant of {a_{1}, ..., a_{m}}∪F), there are h_{1}, …, h_{m} ∈ F and g_{2}, …, g_{m} ∈ F such that

γa_{1}h_{1}, g_{2}γa_{2}h_{2}, …, g_{m}γa_{m}h_{m}

form a ping-pong m-tuple of (\frac{r}{c}, c^{3}ϵ)-very proximal transformations and are free generators of a free group.

Proof. In this proof, whenever we speak about an ϵ-contracting or (r, ϵ)- proximal transformation, we shall choose and fix a point and a projective hyperplane with respect to which it is ϵ-contracting or (r, ϵ)-proximal and shall refer to them as the attracting point and the repulsive hyperplane of that transformation.

As γ is ϵ-very contracting, γa_{1} is cϵ-very contracting. Now by Proposition 3.2 (iii), we can find h_{1} ∈ F such that γa_{1}h_{1} is (\frac{r}{c}, c^{2}ϵ)-very proximal.

We proceed by induction and suppose that x_{1} = γ_{1}h_{1} and x_{j} = g_{j}γa_{j}h_{j} have been constructed for all j < i (i = 2, ..., m). Let’s construct h_{i} and g_{i}. First note that γa_{i} is cϵ-very contracting. We take h_{i} ∈ F such that h_{i}^{−1} maps the attracting point of (γa_{i})^{−1} at a distance at least r from the repulsive hyperplanes of all transformations x_{j} and x_{j}^{−1} for j < i, and h_{i} maps the attracting point of any x_{j} and x_{j}^{−1} (j < i) at least r away from the repulsive hyperplan of γa_{i}. This is possible since F is an (m, r)-separating set. The second requirement from h_{i} implies that the repulsive hyperplan H_{(γa_{i}h_{i})} = h_{i}^{−1}(H_{(γa_{i})}) of γa_{i}h_{i} is at least r/c away from the attracting points of the x_{j}’s and x_{j}^{−1}’s.

Then pick g_{i} ∈ F which takes the attracting point of γa_{i}h_{i} at a distance at least r from the repulsive hyperplanes of all x_{j}’s and x_{j}^{−1}’s for all j < i and also from the repulsive hyperplane of γa_{i}h_{i}, and that g_{i}^{−1} takes the attracting point of any x_{j} and x_{j}^{−1} (j < i) at least r away from the repulsive hyperplane of (γa_{i}h_{i})^{−1}. This means that the repulsive hyperplane g_{i}(H_{(γa_{i}h_{i})^{−1}}) = H_{(g_{i}γa_{i}h_{i})^{−1}} of g_{i}γa_{i}h_{i} is at least r/c away from the attracting points all x_{j} and x_{j}^{−1}. Additionally we can require that g_{i}^{−1} takes the attracting point of (γa_{i}h_{i})^{−1} at a distance at least r from the repulsive hyperplane of (γa_{i}h_{i})^{−1}.

Then clearly, x_{i} = g_{i}γa_{i}h_{i} is (\frac{r}{c}, c^{3}ϵ)-very proximal and (x_{1}, ..., x_{i}) form a ping-pong i-tuple of (\frac{r}{c}, c^{3}ϵ)-very proximal transformations. □

4. Constructing very proximal elements in Γ

Our aim in this section is to establish an action of Γ, on some projective space over a local field, which has many very proximal elements. Proposition 3.2 tells us how to get very proximal elements out of contracting elements when a separating set is available. We shall show how to construct an action of Γ, on a projective space over a local field, with both ingredients : contracting elements and separating set.

For an algebraic number field there are natural associated local fields - its completions. The following Lemma reduces the general case to this.

Lemma 4.1. Let 𝔾 be a semisimple algebraic group defined over ℚ, and G = 𝔾(ℝ)^{0} be the corresponding connected semisimple Lie group. Let Γ be a finitely generated dense subgroup in G. Then, there exists a number field K ⊂ ℝ and a group homomorphism π : Γ → 𝔾 such that π(Γ) ⊂ 𝔾(K) and π(Γ) is still dense in G.

Proof. Choose a finite set of generators γ_{1}, γ_{2}, …, γ_{k} such that γ_{1}, γ_{2} are close to 1 and (γ_{1}, γ_{2}) belongs to the complement of the proper exponential algebraic set introduced in theorem 2.2. Let Γ = {γ_{1}, ..., γ_{k} | (r_{α})_{α}} be a presentation of Γ. The variety of representations

Hom(Γ, 𝔾) = {(g_{1}, ..., g_{k}) ∈ 𝔾^{k} : ∀α, r_{α}(g_{1}, ..., g_{k}) = 0}

is an algebraic sub-variety of 𝔾^{k} which is defined over ℚ . Its set of algebraic points is therefore dense (for the real topology) in its set of real points. Choosing an algebraic point in Hom(Γ, 𝔾(ℝ)) very close to the original (γ_{1}, ..., γ_{k}), we obtain a representation π of Γ into G such that π(Γ) is contained in 𝔾(\overline{ℚ}) and is still dense in G, as small deformations of γ_{1}, γ_{2} generate dense subgroups. Finally, since it is finitely generated, it is contained in 𝔾(K) for some number field K. □

As explained in the last section, “ε-contraction” is equivalent to “big ratio between the first and the second diagonal entries in the KAK decomposition.”

Lemma 4.2 (Constructing contracting elements). Let K be a number field and 𝔾 be a non-trivial semisimple Zariski-connected algebraic group defined over K. Let R be a finitely generated subring of K and I an infinite subset of 𝔾(R). Then for some completion k of K there is an irreducible rational representation ρ : 𝔾 → 𝕊𝕃(V) defined over k, with dim_{k} V ≥ 2, such that the set

\{\frac{a_{1}(ρ(g))}{a_{2}(ρ(g))} : g ∈ I\}

is unbounded, where a_{1}(ρ(g)) and a_{2}(ρ(g)) are the first and second diagonal terms in a Cartan decomposition of ρ(g) in 𝕊𝕃(V).

Proof. As R is finitely generated, the discrete diagonal embedding of K in its adèle group gives rise to a discrete embedding of R into a product of finitely many places. Explicitly, there is a finite set S of places of K, including all archimedean ones, such that R is contained in the ring of S-integers O_{K}(S). Projecting to the finite product of places corresponding to S, we get a discrete embedding

R ↪ ∏_{ν∈S} K_{ν},

which gives rise to a discrete embedding

𝔾(R) ↪ ∏_{ν∈S} 𝔾(K_{ν}).

The infinite set I is mapped to some unbounded set in the latter product. Hence for some place ν ∈ S, the embedding 𝔾(R) ↪ 𝔾(K_{ν}) sends I to an unbounded set. Take k = K_{v}. Since 𝔾 is also defined over k, there exists a k-rational faithful representation of 𝔾 on some k-vector space V_{0}. Under this representation, the set I is sent to some unbounded set in 𝕊𝕃(V_{0}) which we shall continue to denote by I.

Fix a k-basis of V_{0} and consider the Cartan decomposition in 𝕊𝕃(V_{0}) corresponding to this basis. Recall the notations of section 3. For x ∈ 𝕊𝕃(V_{0}), a_{1}(x) ≥ a_{2}(x) ≥ … ≥ a_{d}(x) > 0 denote the correspond diagonal coefficients in the Cartan decomposition of x in 𝕊𝕃(V_{0}). Since their product is 1, there is some 1 ≤ i < d for which the set \{\frac{a_{i}(g)}{a_{i+1}(g)}\}_{g∈I} is unbounded.

Considering the i’th wedge product representation of the above representation of 𝔾, we obtain a rational representation

ρ_{0} : 𝔾 → 𝕊𝕃( ⋀^{i} V_{0})

defined over k. Looking at the Cartan decomposition in 𝕊𝕃( ⋀^{i} V_{0}) with respect to the basis of ⋀^{i} V_{0} induced by the given basis of V_{0}, we see that a_{1}(ρ_{0}(x)) = a_{1}(x)...a_{i}(x) and a_{2}(ρ_{0}(g)) = a_{1}(x)...a_{i−1}(x)a_{i+1}(x), and hence, under this representation, the set

\{\frac{a_{1}(ρ_{0}(g))}{a_{2}(ρ_{0}(g))}\}_{g∈I}

is unbounded.

The representation ρ_{0} might be reducible. However, as 𝔾(k) is a connected semisimple algebraic group, the k-rational representation ρ_{0} is completely reducible. Let W = ⋀^{i} V_{0} be the representation space. Decompose it into 𝔾(k)-irreducible spaces W = W_{1} ⊕ ... ⊕ W_{q}. Any x = ρ_{0}(g) ∈ ρ_{0}(𝔾(k)) ≤ 𝕊𝕃(W) stabilizes each W_{i}, and we shall write x_{i} ∈ 𝕊𝕃(W_{i}) for its restriction to the subspace W_{i} (note that the determinant of x_{i} is 1 as 𝔾(k) is semisimple).

The space W is endowed with the norm ∥·∥ corresponding to the above choice of a basis of V_{0} (as in section 3) for which a_{1}(g) = ∥g∥ and a_{1}(g)a_{2}(g) = ∥⋀^{2} g∥ for any g ∈ 𝕊𝕃(W). Similarly we can choose a basis and a corresponding norms ∥·∥_{i} for each W_{i} such that

a_{1}(x_{i}) = ∥x_{i}∥_{i} and a_{1}(x_{i})a_{2}(x_{i}) = ∥⋀^{2} x_{i}∥_{i} .

As any two norms on the finite dimensional vector space M_{n}(k), (n = dim(W)) are equivalent, there is c_{1} > 0 with respect to which

\frac{1}{c_{1}} max_{1≤j≤q}\{∥x_{j}∥_{j}\} ≤ ∥x∥ ≤ c_{1} max_{1≤j≤q}\{∥x_{j}∥_{j}\}.

Similarly, for some constant c_{2} > 0

\frac{1}{c_{2}} max_{i,j}\{∥(⋀^{2} x)_{ij}∥_{ij}\} ≤ ∥⋀^{2} x∥ ≤ c_{2} max_{i,j}\{∥(⋀^{2} x)_{ij}∥_{ij}\}.

Replacing I by an infinite subset if necessary, we can find an index i_{0} ≤ q with ∥x∥ ≤ c_{1} ∥x_{i_{0}}∥_{i_{0}} for any x of the form x = ρ_{0}(g), g ∈ I. As {∥ρ_{0}(g)∥}_{g∈I} is unbounded, and each x_{i} belongs to 𝕊𝕃(W_{i}), dim W_{i_{0}} ≥ 2. Moreover, we have

∥⋀^{2} x∥ ≥ \frac{1}{c_{2}} ∥(⋀^{2} x)_{ii}∥_{ii} = \frac{1}{c_{2}} ∥⋀^{2} x_{i}∥_{i} ,

which implies

\frac{a_{1}(x)}{a_{2}(x)} ≤ c_{1}^{2}c_{2} \frac{a_{1}(x_{i_{0}})}{a_{2}(x_{i_{0}})}.

Hence ρ = (ρ_{0})_{|W i_{0}} is the desired rational irreducible representation.

□

Lemma 4.3 (Constructing separating sets). Let m be a positive integer, K a local field, 𝔾 a Zariski-connected k-algebraic group, V a k-vector space, and ρ : 𝔾 → 𝕊𝕃(V) a k-irreducible rational representation. Then for any Zariski-dense subset Ω ⊂ 𝔾(k), there exists a finite subset F ⊂ Ω and a positive real number r > 0, which gives rise under ρ to an (m, r)-separating set (cf. 3.2) on ℙ(V).

Proof. For each γ ∈ Γ, let M_{γ} be the set of all tuples (v_{i}, H_{i})_{i=1}^{2m} of 2m points v_{i} ∈ ℙ(V) and 2m projective hyperplanes H_{i} ⊂ ℙ(V), not necessarily different, such that for some 1 ≤ i, j ≤ 2m, γ · v_{i} ∈ H_{j} or γ^{−1} · v_{i} ∈ H_{i}.

We first claim that

(5)  ⋂_{γ∈Ω} M_{γ} = ∅.

Suppose this were not the case. Then there would exist (v_{i}, H_{i})_{i=1}^{2m} such that

Ω ⊂ ⋃_{1≤i,j≤2m} {γ ∈ 𝔾(k), γ · v_{i} ∈ H_{j} or γ^{−1} · v_{i} ∈ H_{j}} .

The sets {γ ∈ 𝔾(k) : γ · v_{i} ∈ H_{j}} and {γ^{−1} ∈ 𝔾(k) : γ · v_{i} ∈ H_{j}} are clearly Zariski-closed and proper, since 𝔾(k) acts irreducibly on V. As 𝔾(k) is Zariski-connected, this yields a contradiction. Thus we have (5).

Second, as the sets M_{γ} are compact in the appropriate product of grassmannians, there is a finite subset F ⊂ Ω, such that

(6)  ⋂_{γ∈F} M_{γ} = ∅.

The following MaxMin of continuous functions

(7)  max_{γ∈F} ( min_{1≤i,j≤2m}{d(γ · v_{i}, H_{j}), d(γ^{−1} · v_{i}, H_{j})})

depends continuously on (v_{i}, H_{i})_{i=1}^{2m}, and by (6) never vanishes. Finally, the compactness of the set of all tuples (v_{i}, H_{i})_{i=1}^{2m} in

(ℙ(V) × 𝔾r_{dim(V)−1}(V))^{2m}

implies that (7) attains a positive minimum r. Thus F gives rise to the desired (m, r)-separating set. □

Putting together Proposition 3.2 with Lemmas 4.2 and 4.3, we obtain a great liberty in the choice of very contracting (or very proximal) elements in Γ.

Proposition 4.1. Let m be a positive integer, K a number field, 𝔾 a Zariskiconnected semisimple K-algebraic group, R ⊂ K a finitely generated subring, I ⊂ 𝔾(R) an infinite subset, and Ω ⊂ 𝔾(K) a Zariski-dense subset of 𝔾. Then there exist:

• An embedding σ : K ↪ k of K into some local field k.

• An irreducible rational representation ρ : 𝔾 → 𝕊𝕃(V) defined over k,

• A finite (m, r)-separating set F ⊂ Ω (for some r > 0) for the action

induced by ρ on the projective space ℙ(V), such that for any ε > 0, there is g ∈ I and f ∈ F for which gfg^{−1} acts as

an ε-very contracting transformation on ℙ(V).

Proof. Lemma (4.2) yields the local field k, the embedding K ↪ k, and the representation ρ, while Lemma (4.3) yields the r-separating set F ⊂ Ω.

For any ε > 0, we can find g ∈ I with

\frac{a_{1}(ρ(g))}{a_{2}(ρ(g))} ≥ \frac{r}{ϵ\sqrt{2Cd}}

where C = max{biLip(f) : f ∈ F } and d is as in Proposition (3.2) (= 4 in archimedean case, \frac{1}{|π|} in the non-archimedean case). By Lemma (3.1), g acts

as a \frac{ϵ\sqrt{2Cd}}{r}-contracting transformation on ℙ(V). Finally, for ϵ small enough, Proposition (3.2) provides f ∈ F such that gfg^{−1} is ϵ-very contracting.

□

Remark 4.1. Note that the original statement of Tits’ alternative ([12], Theorem 1) can be easily derived from the last proposition, first by taking a homomorphism of Γ into 𝔾𝕃_{n}(K), for some number field K, whose image is not almost solvable (this is possible as the index of the solvable subgroup of an almost solvable group and the length of the derived series of solvable groups are both uniformly bounded for groups contained in 𝔾𝕃_{n}) and then projecting to a semisimple factor of the Zariski closure. Now letting Ω = I = Γ in the last Proposition and applying Lemma 3.7 we obtain ping-pong pair.

5. Proofs of theorems 1.1, 1.2, and 1.3

To complete the proofs of the main results of this paper, we shall need some preliminary lemmas.

Lemma 5.1. Let G be a connected real Lie group and Γ ⊂ G a dense subgroup. Then Γ is generated by Γ ∩ U, for any identity neighborhood U in G.

Proof. Let H be the subgroup of G generated by Γ ∩ U. Since Γ is dense and G connected, H is again dense in G. If γ ∈ Γ we can thus find h ∈ H such that h ∈ γU. But then γ^{−1}h ∈ Γ ∩ U ⊂ H. Hence Γ = H. □

Lemma 5.2. Let 𝔾 be a Zariski-connected algebraic group defined over ℝ and let G = 𝔾(ℝ)^{0} be the corresponding connected real Lie group. Let Γ be a dense subgroup in G and π : Γ → G a group homomorphism such that π(Γ) is Zariski-dense. Then for any identity neighborhood U ⊂ G, π(U ∩ Γ) is Zariski-dense in 𝔾.

Proof. Let (U_{n})_{n}, U_{n} ⊃ U_{n+1}, be a nested sequence of identity neighborhoods in G, such that for any identity neighborhood U, U_{n} ⊂ U for large n’s. Let U′_{n} = U_{n} ∩ Γ and let X_{n} be the Zariski closure of π(U′_{n}) in G. Since (X_{n})_{n} is a decreasing sequence of Zariski-closed subsets of G, it stabilizes at some finite step n_{0}. For any g ∈ U′_{n_{0}}, gU′_{n} ⊂ U′_{n_{0}}, for n large enough, hence π(g)X_{n_{0}} = π(g)X_{n} ⊂ X_{n_{0}}. By Lemma 5.1, U′_{n_{0}} generates Γ, hence π(Γ)X_{n_{0}} ⊂ X_{n_{0}}. Since π(Γ) is Zariski dense, this implies X_{n_{0}} = G. Thus π(U ∩ Γ) is Zariski-dense for any identity neighborhood U ⊂ G. □

Proof of theorem 1.1. The adjoint group H = Ad(G) is a center free connected semisimple Lie group, hence is of the form ℍ(ℝ)^{0} where ℍ is some Zariski-connected semisimple ℚ-algebraic group.

By Corollary 2.3, we can assume that Γ is finitely generated. By Lemma 4.1, there is a homomorphism π of Γ into Ad(G) with dense, hence Zariskidense, image which lies in ℍ(K) for some number field K. Since Γ is finitely generated, there is a finitely generated subring R of K, such that π(Γ) ⊂ ℍ(R).

Consider two elements a and b in Γ and a small enough identity neighborhood U′ in G such that for any x ∈ V (a) := U′aU′ and y ∈ V (b) := U′bU′, ⟨x, y⟩ is a dense subgroup of G. By Theorem 2.2 this is always possible. Let U be a smaller identity neighborhood in G such that U^{5} ⊂ U′ and denote

Ω = I = π(Γ ∩ U).

By Lemma 5.2, the conditions of Proposition 4.1 are fulfilled. So we have an action of Γ by projective transformations on some projective space ℙ(V) over a local field k, a finite set F ⊂ Γ ∩ U which acts as a (2, r)-separating set on ℙ(V) (for some r > 0), and for every small ϵ, an element γ_{ϵ} in Γ ∩ U^{4} which acts as an ϵ-very contracting transformation on ℙ(V).

Then by Lemma 3.7, we can find, for every small ϵ > 0, some f_{ϵ}, g_{ϵ}, h_{ϵ} ∈ F such that a_{ϵ} := γ_{ϵ}af_{ϵ} and b_{ϵ} := g_{ϵ}γ_{ϵ}bh_{ϵ} act both as (\frac{r}{c}, c^{2}ϵ)-very proximal transformations on ℙ(V), where c is a constant greater than the maximal bi-Lipschitz constant of {a, b} ∪ F. By Lemma 3.7, they form a ping-pong pair whenever ϵ < \frac{r}{2c^{4}} . Moreover a_{ϵ} ∈ V (a) and b_{ϵ} ∈ V (b). The subgroup ⟨a_{ϵ}, b_{ϵ}⟩ generated by a_{ϵ} and b_{ϵ} is therefore free and dense in G. This concludes the proof. □

The proof of Theorem 1.2 is basically the same as the proof of Theorem 1.1 and we shall not go over it again, but only describe the needed modifications in the above argument.

Proof of Theorem 1.2 (outline). Pick a_{1}, …, a_{m} ∈ Γ close to the identity, and a small identity neighborhood U such that for any selection x_{i} ∈ U^{5}a_{i}U^{5} , i = 1, …, m, the group ⟨x_{1}, …, x_{m}⟩ is dense in G. The existence of such a_{i}’s and U is an easy consequence of Theorem 2.1.

Now project G on the semisimple quotient S = G/R where R is the radical of G, and find a projective space over a local field and an action of Γ (via S) for which there is

• a (m, r)-separating set F ⊂ U ∩ Γ (for some r > 0), and

• for ϵ < \frac{r}{2c^{4}} , (where c is the maximal bi-Lipschitz constant of \{a_{1}, ..., a_{m}\}∪ F when acting on the projective space),

• an element γ_{ϵ} ⊂ U^{4} ∩ Γ acting as an ϵ-very contracting element. Then there are

h_{i} ∈ F (1 ≤ i ≤ m) and g_{i} ∈ F (2 ≤ i ≤ m),

such that

⟨γ_{ϵ}a_{1}h_{1}, g_{2}γ_{ϵ}a_{2}h_{2}, …, g_{m}γ_{ϵ}a_{m}h_{m}⟩

is dense and free. □

Proof of theorem 1.3. Recall that G is a connected non-solvable Lie group and Γ ≤ G is a finitely generated dense subgroup. Define G_{0} = G and G_{n} = \overline{[G_{n−1}, G_{n−1}]}. Then as dim(G) < ∞ the sequence G_{n} stabilizes at some finite step k, H = G_{k} is a connected topologically perfect closed normal subgroup of G, and the quotient G/H is solvable. Moreover it follows by induction on n that Γ ∩ G_{n} is dense in G_{n}.

Let l be the codimension of H and let m be the minimal number of generators of the Lie algebra of H, then l+m ≤ dim(G). Since the projection of Γ to G/H is finitely generated and dense, Proposition 2.5 implies that it contains 2l elements which generate a dense subgroup in G/H. Let a_{1}, …, a_{2l} ∈ Γ be arbitrary liftings of these elements. Additionally, as in the proof of Theorem 1.2, let a_{2l+1}, …, a_{2l+m} be m elements in H ∩ Γ near the identity, and let U ⊂ H be a small identity neighborhood of H, such that for any selection x_{i} ∈ U^{5}a_{i}U^{5} (2l + 1 ≤ i ≤ 2l + m), the group ⟨x_{2l+1}, …, x_{2l+m}⟩ is dense in H. Then for any selection x_{i} ∈ U^{5}a_{i}U^{5} (1 ≤ i ≤ 2l + m) the group ⟨x_{1}, …, x_{2l+m}⟩ is dense in G.

From this point, we can continue exactly as in the proof of Theorem 1.2 in order to find such x_{i}’s such that ⟨x_{1}, …, x_{2l+m}⟩ is free. □

Remark 5.1. The bound 2 dim(G) is not sharp as the latter proof shows. But 2l+m can also be decreased in some cases. For example, it is easy to see that if G is a direct product of a semisimple group with ℝ^{d}, then max{2, 2d} is a sharp bound for Theorem 1.3.

Acknowledgments 5.1. We would like to thank A. Lubotzky and G.A. Margulis for their help and support while working on this question, and also Y. Benoist and S. Mozes for many valuable discussions and comments.

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