Geodesic
Positive crossratios, barycenters, trees and applications to maximal representations
Marc Burger1 ; Alessandra Iozzi1 orcid ; Anne Parreau2 orcid ; Maria Beatrice Pozzetti3 orcid
1ETH Zürich, Switzerland
2Université Grenoble Alpes, France
3Heidelberg University, Germany
Groups, geometry, and dynamics, Tome 18 (2024) no. 3, pp. 799-847

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DOI

We study metric properties of maximal framed representations of fundamental groups of surfaces in symplectic groups over real closed fields, interpreted as actions on Bruhat–Tits buildings endowed with adapted Finsler norms. We prove that the translation length can be computed as intersection with a geodesic current, give sufficient conditions guaranteeing that such a current is a multicurve, and, if the current is a measured lamination, construct an isometric embedding of the associated tree in the building. These results are obtained as application of more general results of independent interest on positive crossratios and actions with compatible barycenters.
DOI : 10.4171/ggd/767
Classification : 22E40, 20F65, 22F05, 57K20
Mots-clés : higher Teichmüller theory, cross ratios, geodesic current, multi curve, maximal representation

Marc Burger  1   ; Alessandra Iozzi  1   ; Anne Parreau  2   ; Maria Beatrice Pozzetti  3

1 ETH Zürich, Switzerland
2 Université Grenoble Alpes, France
3 Heidelberg University, Germany 
Marc Burger; Alessandra Iozzi; Anne Parreau; Maria Beatrice Pozzetti. Positive crossratios, barycenters, trees and applications to maximal representations. Groups, geometry, and dynamics, Tome 18 (2024) no. 3, pp. 799-847. doi: 10.4171/ggd/767
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     title = {Positive crossratios, barycenters, trees and applications to~maximal representations},
     journal = {Groups, geometry, and dynamics},
     pages = {799--847},
     year = {2024},
     volume = {18},
     number = {3},
     doi = {10.4171/ggd/767},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/767/}
}
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