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Preprint

2016-10-13

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.

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Olivier Guichard, Anna Wienhard (2011)

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Quentin Mérigot, Thierry Barbot (2012)