Zariski-dense non-tempered subgroups in higher rank of nearly optimal growth

We call a discrete subgroup Γtempered in 𝐺 if its quasi-regular representation L 2 ⁢ ( Γ \ G ) is tempered.

For each n ≥ 3 , there exists a Zariski-dense, non-lattice, non-tempered subgroup of SO ⁡ ( n , 2 ) .

For a geometrically finite discrete subgroup Γ < SO ⁡ ( n , 1 ) , the hyperbolic manifold Γ \ H n possesses a square-integrable base eigenfunction of the Laplacian if and only if Γ is non-tempered [32, 38, 37]. By contrast, a recent result of [15] shows that, for any non-lattice discrete subgroup Γ of a higher rank simple algebraic group 𝐺, the base eigenfunction on the corresponding locally symmetric manifold is never square-integrable. Hence the appearance of a non-tempered subgroup in Theorem 1.2 underscores another sharp distinction in the behavior of infinite-volume locally symmetric manifolds between the higher rank and rank-one cases.

A discrete subgroup Γ < G has slow growth if ψ Γ ≤ ρ on a + .

Let n ≥ 3 and let Γ be the fundamental group of a closed hyperbolic 𝑛-manifold with a properly embedded totally geodesic hyperplane. For any ε > 0 , there exists a non-empty open subset O = O ⁢ ( ε ) of Hom ⁡ ( Γ , SO ∘ ⁡ ( n , 2 ) ) such that, for any σ ∈ O , the following hold:

1. σ ⁢ ( Γ ) is a Zariski-dense, { α 1 } -Anosov^[1], and non-tempered subgroup of SO ∘ ⁡ ( n , 2 ) without slow growth;

2. for all v ∈ a + , we have

   ψ σ ⁢ ( Γ ) ⁢ ( v ) ≤ ( 2 ⁢ ( n − 1 ) n + ε ) ⁢ ρ ⁢ ( v ) ;

3. there exists a unit vector v σ ∈ a + such that

   (1.3) ψ σ ⁢ ( Γ ) ⁢ ( v σ ) ≥ ( 2 ⁢ ( n − 1 ) n − ε ) ⁢ ρ ⁢ ( v σ ) .

where the supremum is taken over all non-lattice discrete subgroups Λ < SO ∘ ⁡ ( n , 2 ) .

There are many examples of Zariski-dense discrete subgroups in higher rank that are tempered, for instance, the image of any Hitchin representation of a surface group into a real split simple algebraic group of higher rank [16, 12].

Let Γ n be a sequence of discrete subgroups of 𝐺 converging to a closed subgroup 𝐻 in the Chabauty topology. Then 𝐻 is unimodular, and for any non-negative function s ∈ C c ⁢ ( G ) with s ⁢ ( e ) > 0 , we have

Proof

Consider a non-negative function s ∈ C c ⁢ ( G ) with s ⁢ ( e ) > 0 . For simplicity, set ν n = ν Γ n and ν n ′ : = ν n ( s ) − 1 ν n . Then ν n ′ ⁢ ( s ) = 1 .

First we show that the sequence ν n ′ is relatively compact in M ⁢ ( G ) . Since s ⁢ ( e ) > 0 , it follows from the continuity of 𝑠 that there exists a symmetric neighborhood 𝑈 of 𝑒 such that κ : = inf g ∈ U 2 s ( g ) > 0 . Fix any compact subset 𝐶 of 𝐺. Let

Note that

For any n ∈ N , choose a maximal subset F n ⊂ Γ n ∩ C such that g 1 ⁢ U ∩ g 2 ⁢ U = ∅ for all g 1 ≠ g 2 ∈ F n . Then Γ n ∩ C ⊂ F n ⁢ U 2 , so

Therefore, for all n ∈ N , we have

Since 𝐶 is an arbitrary compact subset of 𝐺, it follows that the sequence ν n ′ , n ∈ N , forms a relatively compact subset of M ⁢ ( G ) .

Let ν ∈ M ⁢ ( G ) be a weak-* limit of the sequence ν n ′ . By construction, 𝜈 is a locally finite measure supported on 𝐻 and ν ⁢ ( s ) = 1 . It remains to show that 𝜈 is a Haar measure on 𝐻. Let φ ∈ C c ⁢ ( G ) and h ∈ H . Let γ n ∈ Γ n be a sequence with lim n → ∞ γ n = h . Then, since ν n ′ is a Haar measure of Γ n , we get

The first and the third term converge to zero since ν n ′ weakly converges to 𝜈. The middle term goes to zero because φ ( γ n ⋅ ) converges uniformly to φ ⁢ ( ⋅ ) . Hence, the right-hand side converges to 0 as n → ∞ , so 𝜈 is indeed left 𝐻-invariant. Similarly, we can show that 𝜈 is also a right 𝐻-invariant. This proves that 𝐻 is unimodular. Since ν ⁢ ( s ) = 1 , we have ν = ν H ⁢ ( s ) − 1 ⁢ ν H and thus the desired convergence (2.1) follows from ν n ′ → ν . ∎

The normalization of measures by the integral of 𝑠 is necessary in the above proposition. For example, if G = SL 2 ⁡ ( F p ⁢ ( ( t ) ) ) and

then, as n → ∞ , Γ n converges to the trivial subgroup { e } in the Chabauty topology, but the sequence ν Γ n of counting measures on Γ n fails to converge on the account of mass near identity blowing up to infinity.

Suppose that 𝐺 has the no-small-subgroup property (e.g., a real Lie group). Let Γ n be a sequence of discrete subgroups of 𝐺 which converges to a discrete subgroup Γ in the Chabauty topology. Then, as n → ∞ ,

where δ γ denotes the Dirac measure at { γ } .

Proof

Let ν n : = ∑ γ ∈ Γ n δ γ and ν : = ∑ γ ∈ Γ δ γ . Let φ ∈ C c ⁢ ( G ) . We need to show that

Let ε > 0 be arbitrary. Fix a compact subset C ⊂ G and φ ∈ C c ⁢ ( G ) supported on 𝐶. Enlarging 𝐶 if needed, we may assume that Γ ∩ ∂ C = ∅ . By the hypothesis that 𝐺 has the no-small-subgroup property, there is an open neighborhood U ⊂ G of the identity 𝑒 which contains no non-trivial subgroup of 𝐺. We choose an open symmetric neighborhood U 1 ⊂ G of 𝑒 such that

1. U 1 2 ⊂ U ;

2. γ ⁢ U 1 5 ⊂ C for all γ ∈ Γ ∩ C ;

3. the collection γ ⁢ U 1 5 , γ ∈ Γ ∩ C , are pairwise disjoint;

4. for all γ ∈ C ∩ Γ and u ∈ U 1 ,

   | φ ⁢ ( γ ) − φ ⁢ ( γ ⁢ u ) | ≤ ε # ⁢ ( Γ ∩ C ) .

Note that Γ ∩ C 1 = ∅ . Since the sequence Γ n converges to Γ in the Chabauty topology, we have Γ n ∩ C 1 = ∅ for all 𝑛 large enough. For each fixed γ ∈ Γ ∩ C , there exists n 0 = n 0 ⁢ ( γ ) ≥ 1 such that Γ n ∩ γ ⁢ U 1 ≠ ∅ and Γ n ∩ C 1 = ∅ for all n ≥ n 0 . Since Γ ∩ C is finite, we have n 0 : = max { n 0 ( γ ) : γ ∈ Γ ∩ C } < ∞ .

On the other hand, we claim that, for any γ ∈ C ∩ Γ and n ≥ 1 , # ⁢ ( Γ n ∩ γ ⁢ U 1 ) ≤ 1 . Indeed, suppose there exists some element γ n ∈ Γ n ∩ γ ⁢ U 1 . Then

By the no-small-subgroup property of 𝐺, we have either Γ n ∩ U 1 2 = { e } or there is some element γ n ′ ∈ Γ n ∩ ( U 1 4 \ U 1 2 ) ; otherwise, Γ n ∩ U 1 2 would be a non-trivial subgroup. In the second case, we would have

Using property (3), we get γ n ⁢ γ n ′ ∈ C 1 , contradicting the fact that Γ n ∩ C 1 = ∅ . Therefore, we must have Γ n ∩ U 1 2 = { e } . This implies that Γ n ∩ γ ⁢ U 1 = { γ n } , proving the claim.

Therefore, for all γ ∈ Γ ∩ C and n ≥ n 0 , we have a unique element γ n ∈ Γ n such that Γ n ∩ γ ⁢ U 1 = { γ n } , and γ n → γ as n → ∞ . Since

we get from (4) that, for all n ≥ n 0 ,

This finishes the proof. ∎

Let Γ n be a sequence of discrete subgroups of 𝐺 which converges to a closed unimodular subgroup 𝐻 in the Chabauty topology. Let K < G be a compact subgroup of 𝐺. For any vectors v , w ∈ L 2 ⁢ ( H \ G ) , there exist sequences v n , w n ∈ L 2 ⁢ ( Γ n \ G ) , n ∈ N , such that

1. for all g ∈ G ,

   lim n → ∞ ⟨ v n , g . w n ⟩ L 2 ⁢ ( Γ n \ G ) = ⟨ v , g . w ⟩ L 2 ⁢ ( H \ G ) ,

   and the convergence is uniform on compact subsets of 𝐺;

2. we have

   lim n → ∞ ∥ v n ∥ L 2 ⁢ ( Γ n \ G ) = ∥ v ∥ L 2 ⁢ ( H \ G ) & lim n → ∞ ∥ w n ∥ L 2 ⁢ ( Γ n \ G ) = ∥ w ∥ L 2 ⁢ ( H \ G ) ;

3. we have that, for all n ∈ N ,

   dim ⟨ K . v n ⟩ ≤ dim ⟨ K . v ⟩ and dim ⟨ K . w n ⟩ ≤ dim ⟨ K . w ⟩ .

Proof

Since C c ⁢ ( H \ G ) is dense in L 2 ⁢ ( H \ G ) , the matrix coefficient

can be approximated by the matrix coefficients for continuous compactly supported functions, uniformly on compact subsets of 𝐺. This approximation can be done without increasing the dimensions of the spaces spanned by the 𝐾-orbits of 𝑣 and 𝑤. In fact, let u m be a sequence of compactly supported right 𝐾-invariant functions on H \ G converging to the constant function 1 uniformly on compact subsets of H \ G . Since the multiplication by u m is 𝐾-equivariant, we have dim ⟨ K . ( u m v ) ⟩ ≤ dim ⟨ K . v ⟩ , similarly for 𝑤. The matrix coefficients g ↦ ⟨ u m v , g . u m w ⟩ L 2 ⁢ ( H \ G ) converge to g ↦ ⟨ v , g . w ⟩ L 2 ⁢ ( H \ G ) uniformly on 𝐺. Thus we have shown that v , w can be replaced by compactly supported functions, spanning 𝐾-invariant subspaces of equal or smaller dimension. We need one more step to replace them by continuous functions.

Let ϕ ̃ m ∈ C c ⁢ ( G ) be a sequence of non-negative continuous functions with

and support contained in some neighborhood U ̃ m of 𝑒 such that U ̃ m → { e } as m → ∞ . Define ϕ m ∈ C c ⁢ ( G ) by

where d ⁢ k is the probability Haar measure on 𝐾. Clearly, ϕ m is non-negative, continuous and ∫ ϕ m ⁢ ( g ) ⁢ d g = 1 . The support of ϕ m is contained in U m : = { k U ̃ m k − 1 : k ∈ K } . Note that U m → { e } as m → ∞ ; otherwise, we have, by passing to a subsequence, k m ⁢ g m ⁢ k m − 1 → g for some k m ∈ K converging to k 0 ∈ K , g m ∈ U ̃ m and g ≠ e . Since g m → e as m → ∞ , this is a contradiction.

Consider the convolution v ∗ ϕ m ,

and similarly for w ∗ ϕ m . The functions v ∗ ϕ m and w ∗ ϕ m are continuous compactly supported functions on H \ G .

Since the sequence ϕ m is an approximate identity, the matrix coefficient

converges to g ↦ ⟨ v , g . w ⟩ L 2 ⁢ ( H \ G ) , uniformly on compact sets. Furthermore, because ϕ m is 𝐾-conjugation invariant, the map v ↦ v ∗ ϕ m commutes with the action of 𝐾, i.e.,

It follows that dim ⟨ K . ( v ∗ ϕ m ) ⟩ ≤ dim ⟨ K . v ⟩ , and similarly for 𝑤. Therefore, we may assume without loss of generality that v , w ∈ C c ⁢ ( H \ G ) .

First, let v ̃ 0 ∈ C ⁢ ( G ) be the lift of 𝑣 to 𝐺, i.e., for all g ∈ G , v ̃ 0 ( g ) : = v ( H g ) . We note that dim ⟨ K . v ⟩ = dim ⟨ K . v ̃ 0 ⟩ .

Now, we choose a right 𝐾-invariant non-negative function φ ∈ C c ⁢ ( G ) such that

Define v ̃ ∈ C c ⁢ ( G ) by v ̃ ( g ) : = φ ( g ) v ̃ 0 ( g ) for all g ∈ G . Then, for each g ∈ G , we have

Moreover, dim ⟨ K . v ̃ ⟩ ≤ dim ⟨ K . v ̃ 0 ⟩ = dim ⟨ K . v ⟩ .

Choose a non-negative function s ∈ C c ⁢ ( G ) such that s ⁢ ( e ) > 0 and

Set α n : = ∑ γ ∈ Γ n s ( γ ) , and define v n ∈ C c ∞ ⁢ ( Γ n \ G ) as follows: for all g ∈ G ,

Then dim ⟨ K . v n ⟩ ≤ dim ⟨ K . v ̃ ⟩ ≤ dim ⟨ K . v ⟩ .

Let w ̃ ∈ C c ⁢ ( G ) and w n ∈ C c ∞ ⁢ ( Γ n \ G ) be functions constructed in the same way for the vector 𝑤. We claim that, for all g ∈ G ,

uniformly on compact subsets of 𝐺. Indeed,

Proposition 2.1 yields the weak-* convergence of measures

It follows that

and the convergence is uniform for all 𝑔 and 𝑥 in a given compact subset of 𝐺. Indeed, for x , g ∈ C , 𝐶 compact, the family of functions w ̃ ( ⋅ x g ) is equicontinuous and supported in a single compact set, so the integrals converge uniformly for any weak-* convergent sequence of measures. Since v ̃ is compactly supported, we get

and the convergence is uniform for all 𝑔 in a given compact subset of 𝐺. Since

this finishes the proof of (1) and (3). Claim (2) follows since the above argument applies when v = w and g = e and hence gives ⟨ v n , v n ⟩ L 2 ⁢ ( Γ n \ G ) → ⟨ v , v ⟩ L 2 ⁢ ( H \ G ) and similarly for w n and 𝑤. ∎

This proposition implies that if Γ n converges to 𝐻 in the Chabauty topology, then L 2 ⁢ ( H \ G ) is weakly contained in ⨁ n = n 0 ∞ L 2 ⁢ ( Γ n \ G ) for all n 0 ≥ 1 .^[3]

Theorem 3.1 ([10])

For a unitary representation ( π , H ) of 𝐺, the following are equivalent:

1. 𝜋 is tempered;

2. 𝜋 is almost L 2 -integrable;

3. for any 𝐾-finite unit vectors v 1 , v 2 ∈ H and any g ∈ G ,

   | ⟨ π ⁢ ( g ) ⁢ v 1 , v 2 ⟩ | ≤ ( dim ⟨ π ⁢ ( K ) ⁢ v 1 ⟩ ⋅ dim ⟨ π ⁢ ( K ) ⁢ v 2 ⟩ ) 1 / 2 ⁢ Ξ G ⁢ ( g ) .

We say that a unimodular subgroup 𝐻 is a tempered subgroup of 𝐺 (or 𝐺-tempered) if the quasi-regular representation L 2 ⁢ ( H \ G ) is a tempered representation of 𝐺.

Lemma 3.3 ([3, Proposition 3.1])

Let 𝐻 be a unimodular closed subgroup of 𝐺. If 𝐻 is 𝐺-tempered, then any unimodular closed subgroup H ′ < H is also 𝐺-tempered.

The Chabauty limit of a sequence of tempered discrete subgroups of 𝐺 is unimodular and tempered.

Proof

Suppose that Γ n is a sequence of tempered discrete subgroups converging to a closed subgroup 𝐻 in the Chabauty topology. We have that 𝐻 is unimodular by Proposition 2.1. We claim that L 2 ⁢ ( H \ G ) is tempered. Suppose not. By Theorem 3.1, there exist 𝐾-finite unit vectors v , w ∈ L 2 ⁢ ( H \ G ) and g ∈ G such that

By Proposition 2.4, there exist vectors v n , w n ∈ L 2 ⁢ ( Γ n \ G ) such that

We can normalize v n , w n to be unit vectors without affecting the above properties. We deduce that, for all 𝑛 large enough,

This is a contradiction since L 2 ⁢ ( Γ n \ G ) is tempered.

Alternatively, one can use [18, Theorem 4.2], which shows that L 2 ⁢ ( H \ G ) is weakly contained in the direct sum ⨁ n = 1 ∞ L 2 ⁢ ( Γ n \ G ) . If Γ n were all tempered, we would deduce that L 2 ⁢ ( H \ G ) is weakly contained in ⨁ n = 1 ∞ L 2 ⁢ ( G ) , hence in L 2 ⁢ ( G ) , which then implies that 𝐻 is tempered. ∎

We say that a sequence of discrete subgroups Γ i of 𝐺 converges to a discrete subgroup Γ algebraically if there exists a sequence of isomorphisms χ i : Γ → Γ i such that, for all γ ∈ Γ , χ i ⁢ ( γ ) converges to 𝛾 as i → ∞ . In other words, χ i converges to the natural inclusion id Γ in Hom ⁡ ( Γ , G ) , where the space Hom ⁡ ( Γ , G ) is endowed with the topology of pointwise convergence. In this case, Γ is called the algebraic limit of Γ i

We refer the readers to [4] for a comparison of algebraic and Chabauty convergence; in particular, each notion fails to imply the other in general.

The algebraic limit of a sequence of tempered discrete subgroups of 𝐺 is tempered.

Proof

Let Γ i be a sequence of tempered discrete subgroups of 𝐺 which converges to a discrete subgroup Γ algebraically. By passing to a subsequence if necessary, we may assume that Γ i converges to a closed subgroup 𝐻 in the Chabauty topology. Since Γ is the algebraic limit of Γ i , we have Γ < H .

By Theorem 3.4, 𝐻 is unimodular and tempered. Since any closed unimodular subgroup of a tempered subgroup is tempered by Lemma 3.3, Γ is tempered as desired. ∎

If a discrete subgroup Γ is a non-tempered subgroup of 𝐺, there exists an open neighborhood 𝒪 of id Γ in Hom ⁡ ( Γ , G ) such that, for any σ ∈ O , σ ⁢ ( Γ ) is non-tempered.