Notes of Pierre Py’s lecture Cambridge 9-02-2016

Cubulable Kahler groups

Joint with Thomas Delzant.

1. Introduction

Kahler group means fundamental group of a compact Kahler manifold, i.e. a complex
manifold admitting a Kahler metric, i.e. a Hermitian metric whose imaginary part is
a closed 2-form. This includes complex projective manifolds. Finding which nitely
presented groups are Kahler, or merely restrictions on Kahler group, is a classical
problem (Serre, Gromov, Simpson, Toledo,…). More recently, Delzant and Gromov
introduced ideas from geometric group theory in the area.
Finding interesting examples is also part of the problem : Toledo, Dimca-Papadima-
Suciu, Llosa Isenrich, Bridson,…
Today, I will explain restrictions on actions of Kahler groups on finite dimensional
$CAT(0)$ cubical complexes. This is to be expected: as Sageev pointed out, such
actions are related to codimension 1 subgroups ${H < G}$, especially to the number of
ends ${e(G;H)}$ of the relative Schreier graph. Gromov observed that innite Kahler
groups are 1-ended. Napier and Ramachandran showed that if a Kahler group G
has a subgroup H with ${e(G;H) > 2}$, then G virtually surjects onto a surface group.
Delzant and Gromov expanded this: if a Kahler group has “many” subgroups with
${> 2}$ relative ends, then G embeds in a direct product of surface groups. This lead
Dimca-Papadima-Suciu to investigate which subgroups of products of surface groups
are Kahler.

2. Results

Say a group is cubulable if it properly discontinuously on a nite dimensional ${CAT(0)}$
cubical complexes.

Theorem 1 A cubulable Kahler group is virtually a direct product of copies of ${{\mathbb Z}}$ and surface groups.
Furthermore, the product structure relates to the irreducible decomposition of the ${CAT(0)}$
cubical complex ${Y}$ on which ${G}$ acts essentially (i.e. orbits not contained in a bounded
neighborhood of a hyperplane). The action is a product action.

What can one say about Kahler manifolds whose fundamental group are cubulable?

Theorem 2 Let ${X}$ be a closed projective manifold. Assume that X has the same homotopy type as ${Y/G}$ where ${Y}$ is a ${CAT(0)}$ cubical complex and G acts freely co- compactly properly discontinuously on Y . Then X has a finite cover which is biholo- morphic to a product of a complex torus and Riemann surfaces.

The conclusion in the more general Kahler case is slightly weaker.

3. Ingredients

Caprace-Sageev?s work on ${CAT(0)}$ cubical complexes. Bridson-Howie-Miller-Short?s work on subgroups of direct products of surface groups or free groups: ${H}$ can be ${FP^n}$ only if ${H}$ itself is a direct product of subgroups of factors.

4. One more result

4.1. The case of irreducible cube complexes

Theorem 3 Assume that a Kahler group ${G}$ acts on an irreducible ${CAT(0)}$ cubical complex ${Y}$ which is locally finite. Assume that

• the G-action is essential ;
• no invariant flats ;
• no fixed points in the visual boundary.

Then ${G}$ virtually surjects onto a surface group, ${Y}$ contains a convex ${G}$-invariant set ${C}$ on which the action virtually factors trough a surface group.

We would like to remove the local finiteness assumption on ${Y}$, but we are unable to do so now. If we could do so, we could remove the no fixed point assumption as well, thanks to results by Caprace-Chatterji-Fernos.

Theorems 1 and 2 follow from Theorem 3.
Harmonic maps are of no help, since the needed vanishing theorem is not available for cube complex targets.

6. Proof

We shall merely use harmonic functions.
Let ${G = \pi_1(X)}$ act on ${Y}$. If ${h \subset Y}$ is a half-space, we denote by ${\hat h}$ the corresponding hyperplane. Let ${G_{\hat h}}$ denote the subgroup which preserves ${\hat h}$ and each of the compo- nents of ${Y \ \hat h}$.
Fix a half-space ${h}$. Let ${v_h}$ denote the signed ${CAT(0)}$ distance to the hyperplane ${\hat h}$. Let ${f : \tilde X \rightarrow Y}$ be an arbitrary continuous ${G}$-equivariant map. Define

$\displaystyle w_h := v_h ? f : G_{\hat h} \setminus \tilde X \rightarrow {\mathbb R}.$

If ${Y}$ is locally finite, we can show that ${w_h}$ is proper, so ${G_{\hat h} \setminus\tilde X}$ has at least 2 ends. The set of ends splits in two subsets according to the sign of ${w_h}$. We produce a harmonic function tending to 1 (resp. ?1) in such ends.
The following can be found in Kapovich?s paper on Gromov?s proof of Stallings? theorem.

Theorem 4 (Li-Tam, Woess-Kaimanovich, Ramachandran-Kapovich) Let ${M}$ be a bounded geometry Riemannian manifold. Assume ${M}$ satisfies a linear isoperimetric inequality. Then any continuous function

$\displaystyle Ends(M) ? \{-1,1\}$

has a continuous harmonic extension to ${M}$ of finite energy.

Under the assumptions of Theorem 3, one can pick ${h}$ such that ${G}$ contains a non-abelian free group ${F < G}$ such that

$\displaystyle \forall g\in G, F\cap gG_{\hat h}g^{-1}=\{1\}.$

We deduce that ${G_{\hat h} \setminus G}$ satisfies a linear isoperimetric inequality (otherwise, ${F}$ would
have almost invariant vectors on ${L^2(G_{\hat h} \setminus G)}$, which is not true).
Now we have a harmonic proper finite energy function ${u_h : G_{\hat h} \setminus \tilde X \rightarrow {\mathbb R}}$. Stokes and
a cut-off argument imply that ${u_h}$ is pluriharmonic.
We must show that the associated (singular) foliation is in fact a fibration. The point
is to find a psh function ${v}$ which is not a function of ${u_h}$.
Once this is done, we get a proper holomorphic map ${G_{\hat h} \setminus \tilde X \rightarrow}$ a Riemann surface. Let ${H}$ denote the kernel of the corresponding morphism on fundamental groups. We prove that ${H}$ has fixed points on ${Y}$. The trick (due to Behrstock and Charney) is to choose ${\hat h}$ so that its ${G}$-orbit contains 2 strongly separated hyperplanes.