Maximal representations of uniform complex hyperbolic lattices (Q519651)

For lattices in semisimple Lie groups of rank \(1\), rigidity questions are considerably more subtle compared to the case of higher rank lattices. Even among the rank \(1\) groups, there is a lot of variation. For instance, cocompact lattices in \(\mathrm{Sp}(n,1)\) are super-rigid; in particular, they are arithmetic. However, there exist non-arithmetic lattices in \(\mathrm{SO}(n,1)\) for every \(n\). In the case of \(\mathrm{SU}(n,1)\), many things are as yet unknown; non-arithmetic lattices are known to exist for \(n=2,3\) and for no higher \(n\). \noindent Lattices in \(\mathrm{SO}(n,1)\) can sometimes be deformed nontrivially in \(\mathrm{SO}(m,1)\) for \(m>n \geq 3\) as Johnson and Millson showed. However, as demonstrated by \textit{W. M. Goldman} and \textit{J. J. Millson} [Invent. Math. 88, 495--520 (1987; Zbl 0627.22012)], for a cocompact lattice \(\Gamma \subset \mathrm{SU}(n,1)\) with \(n \geq 2\), the representation \(\rho : \Gamma \hookrightarrow \mathrm{SU}(n,1) \hookrightarrow \mathrm{SU}(m,1)\) for \(m \geq n\) is locally rigid even though it may not be infinitesimally rigid. \noindent In this important work, the authors are interested in rigidity properties of representations of cocompact lattices \(\Gamma\) of \(\mathrm{SU}(n,1)\) into semisimple Lie groups of Hermitian type without compact factors. Recall that the classical Hermitian-type noncompact groups are \(\mathrm{SU}(p,q)\) with \(p \geq q \geq 1\), \(\mathrm{SO}_0(p,2)\) with \(p \geq 3\), \(\mathrm{Sp}(m, \mathbb{R})\) with \(m \geq 2\) and \(\mathrm{SO}^{\ast}(2m)\) with \(m \geq 4\). In particular, the authors consider certain representations called maximal representations. To define this, consider any representation \(\rho : \Gamma \rightarrow G\) for any semisimple group \(G\) of Hermitian type without compact factors. Let \(\mathcal{Y}\) denote the symmetric space of \(G\) and consider the \(G\)-invariant Kähler form \(\omega_{\mathcal{Y}}\) on \(\mathcal{Y}\) so normalized that its holomorphic sectional curvatures belong to \([-1,-1/\)rank\((\mathcal{Y})]\). Given any \(\rho\)-equivariant map \(f\) from the complex hyperbolic \(n\)-space to \(\mathcal{Y}\) (which exist as \(\mathcal{Y}\) is contractible), one defines the \textit{Toledo invariant of \(\rho\)} as: \[ \tau(\rho) = \frac{1}{ n!} \int_{\Gamma \backslash \mathbb{H}_{\mathbb{C}}} f^{\ast} \omega_{\mathcal{Y}} \wedge \omega^{n-1} \] where \(\omega\) is the Kähler form on \(\Gamma \backslash \mathbb{H}_{\mathbb{C}}\) coming from \(\mathbb{H}_{\mathbb{C}}\) and \(f^{\ast} \omega_{\mathcal{Y}}\) is regarded as a \(2\)-form on \(\Gamma \backslash \mathbb{H}_{\mathbb{C}}\). The Toledo invariant is independent of the choice of \(f\). It is known that the Toledo invariant satisfies the Milnor-Wood type inequality \[ |\tau(\rho)| \leq \mathrm{rank}(\mathcal{Y}) \mathrm{Vol}(\Gamma \backslash \mathbb{H}_{\mathbb{C}}). \] \noindent The special class of representations \(\rho\) for which the above is an equality form the maximal representations. Toledo himself had studied the case \(n=1, G = \mathrm{SU}(m,1)\). For \(n \geq 2\) and \(G\) of rank \(\geq 2\), very little was known until recently. Very recently, M. B. Pozzetti proved that when \(n \geq 2\) and \(G = \mathrm{SU}(p,q)\) with \(p>q>1\), there are no maximal representations with Zariski dense image. Her proof works for non-cocompact lattices but it does require a target of non-tube type. The authors prove: Let \(\Gamma \leq \mathrm{SU}(n,1)\) be a cocompact, torsion-free lattice with \(n \geq 2\). Let \(\rho : \Gamma \rightarrow G\) be a representation where \(G\) is classical, noncompact of Hermitian type. If \(\rho\) is maximal, then \(G\) must be \(\mathrm{SU}(p,q)\) with \(p \geq qn\), \(\rho\) must be reductive and there exists a holomorphic or antiholomorphic \(\rho\)-equivariant map from the complex hyperbolic \(n\)-space to the symmetric space corresponding to \(\mathrm{SU}(p,q)\). \noindent The authors are thus able to completely describe the maximal representations; this also implies the local rigidity of maximal representations. \noindent Interestingly, although earlier work of \textit{M. Burger} et al. [Geom. Funct. Anal. 19, No. 3, 678--721 (2009; Zbl 1188.53050)] shows that maximal representations are reductive, the present authors' method which is quite different also proves it again independently. The authors use the harmonic Higgs bundle on \(\Gamma \backslash \mathbb{H}_{\mathbb{C}}\) associated to \(\rho\). The paper is almost self-contained.