Johnson homomorphisms and actions of higher-rank lattices on right-angled Artin groups. (Q2869834)

The ultimate goal of this paper is about the image of a homomorphism from an irreducible lattice \(\Lambda\) in \(G\), where \(G\) is a real semisimple Lie group with finite centre, no compact factors and \(\mathrm{rank}_{\mathbb R}G\geq 2\), into a right-angled Artin group. But not less relevant are the several other results that the author obtains for RAAGs (right-angled Artin groups). First several results known for the free group, some of them quite recent, are extend for the RAAGs. Perhaps the most known examples of RAAGs are the free Abelian and the free groups of finite rank. In section 3 the author studies the lower central series of an arbitrary RAAG.NEWLINENEWLINE The main result is Theorem 3.11, which is stated in terms of the Lie algebra associated to the lower central series for the group. Let \(A_\Gamma\) be a RAAG where \(\Gamma\) is a certain graph.NEWLINENEWLINE Theorem 3.11: If (the centre) \(Z(A_\Gamma)=\{1\}\), then \(Z(L)=Z(L/pL)=0\) and \(Z(L/(\bigoplus_{i=c+1}^\infty L_i))=L_i/(\bigoplus_{i=c+1}^\infty L_i)\).NEWLINENEWLINE In section 4 the Torelli group \(\mathbb{T}(A_\Gamma)\) is studied. More precisely, the image of the Torelli group on \(\mathrm{Out}(A_\Gamma)\). It is shown that in case \(Z(A_\Gamma)=\{1\}\), then the successive quotients by the image of the lower central series in \(\mathrm{Out}(A_\Gamma)\) are free Abelian groups. Further, the main result for this section is:NEWLINENEWLINE Theorem 4.9: For any graph \(\Gamma\), the group \(\overline{\mathbb T}(A_\Gamma)\) is residually torsion-free nilpotent.NEWLINENEWLINE In section 5 the concept of \(\mathrm{SL}\)-dimension of a graph is defined and denoted by \(d_{\mathrm{SL}}(\Gamma)\). This is used to state the main result Theorem 6.3 which is a little technical. In section 7 there are applications where the main result, which is close to the Theorem 6.3, says:NEWLINENEWLINE Theorem 7.6: Let \(G\) be a real, semisimple Lie group with finite centre, no compact factors and \(\mathrm{rank}_{\mathbb R}G\geq 2\). Let \(\Lambda\) be an irreducible lattice in \(G\). If \(\mathrm{rank}_{\mathbb R}\geq d_{SL}(\Gamma)\), then every homomorphism \(f\colon\Lambda\to\mathrm{Out}(A_\Gamma)\) has finite image.NEWLINENEWLINE Several techniques are used depending on the topic.