P-adic lattices are not K\"ahler groups

In this note we show that any lattice in a simple p-adic Lie group is not the fundamental group of a compact Ka\"hler manifold, as well as some variants of this result.


1.B.
In this note we consider the Kähler problem for lattices in simple groups over local fields. Recall that if G is a locally compact topological group, a subgroup Γ ⊂ G is called a lattice if it is a discrete subgroup of G with finite covolume (for any G-invariant measure on the locally compact group G).
We work in the following setting. Let I be a finite set of indices. For each i ∈ I we fix a local field k i and a simple algebraic group G i defined and isotropic over k i . Let G = i∈I G i (k i ). The topology of the local fields k i , i ∈ I, makes G a locally compact topological group. We define rk G := i∈I rk k i G i .
We consider Γ ⊂ G an irreducible lattice: there does not exist a disjoint decomposition I = I 1 I 2 into two non-empty subsets such that, for j = 1, 2, the subgroup Γ j := Γ ∩ G I j of G I j := i∈I j G i (k i ) is a lattice in G I j .
The reference for a detailed study of such lattices is [Mar91]. In Section 2 we recall a few results for the convenience of the reader.

1.C.
Most of the lattices Γ as in Section 1.B are finitely presented (see Section 2.C). The question whether Γ is Kähler or not has been studied by Simpson using his non-abelian Hodge theory when at least one of the k i 's is archimedean. He shows that if Γ is Kähler then necessarily for any i ∈ I such that k i is archimedean the group G i has to be of Hodge type (i.e. admits a Cartan involution which is an inner automorphism), see [Si92, Corollary 5.3 and 5.4]. For example SL(n, Z) is not a Kähler group as SL(n, R) is not a group of Hodge type. In this note we prove: Theorem 1.1. Let I be a finite set of indices and G be a group of the form j∈I G j (k j ), where G j is a simple algebraic group defined and isotropic over a local field k j . Let Γ ⊂ G be an irreducible lattice.
Suppose there exists an i ∈ I such that k i is non-archimedean. If rk G > 1 and char(k i ) = 0, or if rk G = 1 (i.e. G = G(k) with G a simple isotropic algebraic group of rank 1 over a local field k) then Γ is not a Kähler group.
Remark 1.2. Notice that the case rk G = 1 is essentially folkloric. As we did not find a reference in this generality let us give the proof in this case.
If Γ is not cocompact in G (this is possible only if k has positive characteristic) then Γ is not finitely generated by [L91,Corollary 7.3], hence not Kähler.
Hence we can assume that Γ is cocompact. In that case it follows from [L91, Theorem 6.1 and 7.1] that Γ admits a finite index subgroup Γ which is a (non-trivial) free group. But a non-trivial free group is never Kähler, as it always admits a finite index subgroup with odd Betti number (see [ABCKT96, Example 1.19 p.7]). Hence Γ , thus also Γ , is not Kähler.
On the other hand, to the best of our knowledge not a single case of Theorem 1.1 in the case where rk G > 1 and all the k i , i ∈ I, are non-archimedean fields of characteristic zero was previously known. The proof in this case is a corollary of Margulis' superrigidity theorem and the recent integrality result of Esnault and Groechenig ([EG17, Theorem 1.3], whose proof was greatly simplified in ).

1.D. Let us mention some examples of Theorem 1.1:
-Let p be a prime number, is not a Kähler group. This is new for n ≥ 3.
-I = {1; 2}, k 1 = R and G 1 = SU(r, s) for some r ≥ s > 0, k 2 = Q p and G 2 = SL(r + s). Then any irreducible lattice in SU (r, s) × SL(r + s, Q p ) is not Kähler. In Section 2 we recall how to construct such lattices (they are S-arithmetic). The analogous result that any irreducible lattice in SL(n, R)×SL(n, Q p ) (for example SL(n, Z[1/p])) is not a Kähler group already followed from Simpson's theorem.
1.E. I don't know anything about the case not covered by Theorem 1.1: can a (finitely presented) irreducible lattice in G = i∈I G i (k i ) with rk G > 1 and all k i of (necessarily the same, see Theorem 2.1) positive characteristic, be a Kähler group? This question already appeared in [BKT13, Remark 0.2 (5)].

Reminder on lattices 2.A.
Examples of pairs (G, Γ ) as in Section 1.B are provided by S-arithmetic groups: let K be a global field (i.e a finite extension of Q or F q (t), where F q denotes the finite field with q elements), S a non-empty set of places of K, S ∞ the set of archimedean places of K (or the empty set if K has positive characteristic), O S∪S ∞ the ring of elements of K which are integral at all places not belonging to S ∪ S ∞ and G an absolutely simple K-algebraic group, anisotropic at all archimedean places not belonging to If S is finite the image Γ in v∈S G(K v ) of an S-arithmetic group Λ by the diagonal map is an irreducible lattice (see [B63] in the number field case and [H69] in the function field case). In the situation of Section 1.B, a (necessarily irreducible) lattice Γ ⊂ G is called S-arithmetic if there exist K, G, S as above, a bijection i : S −→ I, isomorphisms K v −→ k i(v) and, via these isomorphisms, k i -isomorphisms ϕ i : G −→ G i such that Γ is commensurable with the image via i∈I ϕ i of an S-arithmetic subgroup of G(K).

2.B.
Margulis' and Venkataramana's arithmeticity theorem states that as soon as rk G is at least 2 then every irreducible lattice in G is of this type: 1 (Margulis, Venkataramana). In the situation of Section 1.B, suppose that Γ ⊂ G is an irreducible lattice and that rk G ≥ 2. Suppose moreover for simplicity that G i , i ∈ I, is absolutely simple. Then: (a) All the fields k i have the same characteristic.
Remark 2.2. Margulis [Mar84] proved Theorem 2.1 when char(k i ) = 0 for all i ∈ I. Venkatarama [V88] had to overcome many technical difficulties in positive characteristics to extend Margulis' strategy to the general case.
On the other hand, if rk G = 1 (hence I = {1}) and k = k 1 is non-archimedean, there exist non-arithmetic lattices in G, see [L91, Theorem A].

2.C.
With the notations of Section 2.A, an S-arithmetic lattice Γ is always finitely presented except if K is a function field, and rk K G = rk G = |S| = 1 (in which case Γ is not even finitely generated) or rk K G > 0 and rk G = 2 (in which case Γ is finitely generated but not finitely presented). In the number field case see the result of Raghunathan [R68] in the classical arithmetic case and of Borel-Serre [BS76] in the general S-arithmetic case; in the function field case see the work of Behr, e.g. [Behr98]. For example the lattice SL 2 (F q [t]) of SL 2 (F q ((1/t))) is not finitely generated, while the lattice SL 3 (F q [t]) of SL 3 (F q ((1/t))) is finitely generated but not finitely presented.

Proof of Theorem 1.1
Thanks to Remark 1.2 we can assume that rk G > 1. In this case the main tools for proving Theorem 1.1 are the recent result of Esnault and Groechenig and Margulis' superrigidity theorem.

3.A.
Recall that a linear representation ρ : Γ −→ GL(n, k) of a group Γ over a field k is cohomologically rigid if H 1 (Γ , Ad ρ) = 0. A representation ρ : Γ −→ GL(n, C) is said to be integral if it factorizes through ρ : Γ −→ GL(n, K), K → C a number field, and moreover stabilizes an O K -lattice in C n (equivalently, see [Ba80, Corollary 2.3 and 2.5]: for any embedding v : K → k of K in a non-archimedean local field k the composed representation ρ v : Γ −→ GL(n, K) → GL(n, k) has bounded image in GL(n, k)). A group will be said complex projective if is isomorphic to the fundamental group of a connected smooth complex projective variety. This is a special case of a Kähler group (the question whether or not any Kähler group is complex projective is open).
In [EG17-2, Theorem 1.1] Esnault and Groechenig prove that if Γ is a complex projective group then any irreducible cohomologically rigid representation ρ : Γ −→ GL(n, C) is integral. This was conjectured by Simpson.

3.B.
A corollary of [EG17-2, Theorem 1.1] is the following: Corollary 3.1. Let Γ be a complex projective group. Let k be a non-archimedean local field of characteristic zero and let ρ : π 1 (X) −→ GL(n, k) be an absolutely irreducible cohomologically rigid representation. Then ρ has bounded image in GL(n, k).
Proof. Let k be an algebraic closure of k. As ρ is absolutely irreducible and cohomologically rigid there exists g ∈ GL(n, k) and a number field K ⊂ k such that ρ g (Γ ) := g · ρ · g −1 (Γ ) ⊂ GL(n, k) lies in GL(n, K).
Let k be the finite extension of k generated by K and the matrix coefficients of g and g −1 . This is still a non-archimedean local field of characteristic zero, and both ρ(Γ ) and ρ g (Γ ) are subgroups of GL(n, k ). As ρ : Γ −→ GL(n, k) ⊂ GL(n, k ) has bounded image in GL(n, k) if and only if ρ g : Γ −→ GL(n, k ) has bounded image in GL(n, k ), we can assume, replacing ρ by ρ g and k by k if necessary, that ρ(Γ ) is contained in GL(n, K) with K ⊂ k a number field.
Let σ : K → C be an infinite place of K and consider ρ σ : Γ ρ −→ GL(n, K) σ → GL(n, C) the associated representation. As ρ is absolutely irreducible, the representation ρ σ is irreducible. As the representation ρ σ is cohomologically rigid.
It follows from [EG17, Theorem 1.3] that ρ σ is integral. In particular, considering the embedding K ⊂ k, it follows that the representation ρ : Γ −→ GL(n, k) has bounded image in GL(n, k).

3.C.
Notice that we can upgrade Corollary 3.1 to the Kähler world if we restrict ourselves to faithful representations: Corollary 3.2. The conclusion of Corollary 3.1 also holds for Γ a Kähler group and ρ : π 1 (X) −→ GL(n, k) a faithful representation.
Proof. As the representation ρ is faithful, the group Γ is a linear group in characteristic zero. It then follows that the Kähler group Γ is a complex projective group (see [CCE14, Theorem 0.2] which proves that a finite index subgroup of Γ is complex projective, and its refinement [C17, Corollary 1.3] which proves that Γ itself is complex projective). The result now follows from Corollary 3.1.

3.D.
Let us apply Corollary 3.1 to the case of Theorem 1.1 where rk G > 1. Renaming the indices of I if necessary, we can assume that I = {1, · · · , r} and k 1 is non-archimedean of characteristic zero. Let us choose an absolutely irreducible k 1 -representation ρ G 1 : be the representation of Γ deduced from ρ G 1 (where p 1 : G −→ G 1 (k 1 ) denotes the projection of G onto its first factor). As p 1 (Γ ) is Zariski-dense in G 1 it follows that ρ is absolutely irreducible.
Suppose by contradiction that Γ is a Kähler group. By Theorem 2.1 (a) and the assumption that k 1 has characteristic zero it follows that Γ is linear in characteristic zero. As in the proof of Corollary 3.2 we deduce that Γ is a complex projective group. It then follows from Corollary 3.1 that ρ has bounded image in GL(V ), hence that p 1 (Γ ) is relatively compact in G(k 1 ). This contradicts the fact that Γ is a lattice in G = G(k 1 ) × j∈I\{1} G(k j ).