7
views
0
recommends
+1 Recommend
0 collections
    0
    shares
      • Record: found
      • Abstract: found
      • Article: found
      Is Open Access

      Pappus theorem, schwartz representations and anosov representations

      Preprint
      , ,

      Read this article at

      Bookmark
          There is no author summary for this article yet. Authors can add summaries to their articles on ScienceOpen to make them more accessible to a non-specialist audience.

          Abstract

          In the paper Pappus's theorem and the modular group [13], R. Schwartz constructed a 2-dimensional family of faithful representations \(\rho\)$\Theta\( of the modular group PSL(2, Z) into the group G of projective symmetries of the projective plane via Pappus Theorem. If PSL(2, Z)o denotes the unique index 2 subgroup of PSL(2, Z) and PGL(3, R) the subgroup of G consisting of projective transformations, then the image of PSL(2, Z)o under \)\rho\(\)\Theta\( is in PGL(3, R). The representations \)\rho\(\)\Theta\( share a very interesting property with Anosov representations of surface groups into PGL(3, R): It preserves a topological circle in the flag variety. However, the representation \)\rho\(\)\Theta\( itself cannot be Anosov since the Gromov boundary of PSL(2, Z) is a Cantor set and not a circle. In her PhD Thesis [15], V. P. Val{\'e}rio elucidated the Anosov-like feature of the Schwartz representations by showing that for each representation \)\rho\(\)\Theta\(, there exists an 1-dimensional family of representations (\)\rho\( \)\epsilon\( \)\Theta\() \)\epsilon\(\)\in\(R of PSL(2, Z)o into PGL(3, R) such that \)\rho\( 0 \)\Theta\( is the restriction of the Schwartz representation \)\rho\(\)\Theta\( to PSL(2, Z)o and \)\rho\( \)\epsilon\( \)\Theta\( is Anosov for every \)\epsilon\( \textless{} 0. This result was announced and presented in her paper [14]. In the present paper, we extend and improve her work. For every representation \)\rho\(\)\Theta\(, we build a 2-dimensional family of representations (\)\rho\( \)\lambda\( \)\Theta\() \)\lambda\(\)\in\(R 2 of PSL(2, Z)o into PGL(3, R) such that \)\rho\( \)\lambda\( \)\Theta\( = \)\rho\( \)\epsilon\( \)\Theta\( for \)\lambda\( = (\)\epsilon\(, 0) and \)\rho\( \)\lambda\( \)\Theta\( is Anosov for every \)\lambda\( \)\in\( R \)\bullet\( , where R \)\bullet\( is an open set of R 2 containing {(\)\epsilon\(, 0) | \)\epsilon$ \textless{} 0}. Moreover, among the 2-dimensional family of new Anosov representations, an 1-dimensional subfamily of representations can extend to representations of PSL(2, Z) into G, and therefore the Schwartz representations are, in a sense, on the boundary of the Anosov representations in the space of all representations of PSL(2, Z) into G.

          Related collections

          Most cited references8

          • Record: found
          • Abstract: not found
          • Book Chapter: not found

          Hyperbolic Groups

          M. Gromov (1987)
            Bookmark
            • Record: found
            • Abstract: found
            • Article: found
            Is Open Access

            Anosov representations: Domains of discontinuity and applications

            The notion of Anosov representations has been introduced by Labourie in his study of the Hitchin component for SL(n,R). Subsequently, Anosov representations have been studied mainly for surface groups, in particular in the context of higher Teichmueller spaces, and for lattices in SO(1,n). In this article we extend the notion of Anosov representations to representations of arbitrary word hyperbolic groups and start the systematic study of their geometric properties. In particular, given an Anosov representation of \(\Gamma\) into G we explicitly construct open subsets of compact G-spaces, on which \(\Gamma\) acts properly discontinuously and with compact quotient. As a consequence we show that higher Teichmueller spaces parametrize locally homogeneous geometric structures on compact manifolds. We also obtain applications regarding (non-standard) compact Clifford-Klein forms and compactifications of locally symmetric spaces of infinite volume.
              Bookmark
              • Record: found
              • Abstract: not found
              • Article: not found

              Anosov AdS representations are quasi-Fuchsian

                Bookmark

                Author and article information

                Journal
                2016-10-13
                Article
                1610.04049
                b6982303-0900-4b9e-84e7-241cd3c6af7e

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                History
                Custom metadata
                math.DS math.RT
                ccsd

                Differential equations & Dynamical systems,Algebra
                Differential equations & Dynamical systems, Algebra

                Comments

                Comment on this article