Non-distortion of twin building lattices

Caprace, Pierre-Emmanuel;Remy, Bertrand
(2010) Geometriae Dedicata — Vol. 147, n° 1, p. 397-408 (2010)

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Abstract
We show that twin building lattices are undistorted in their ambient group; equivalently, the orbit map of the lattice to the product of the associated twin buildings is a quasi-isometric embedding. As a consequence, we provide an estimate of the quasi-flat rank of these lattices, which implies that there are infinitely many quasi-isometry classes of finitely presented simple groups. Finally, we describe how non-distortion of lattices is related to the integrability of the structural cocycle.
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Caprace, P.-E., & Remy, B. (2010). Non-distortion of twin building lattices. Geometriae Dedicata, 147(1), 397-408. https://doi.org/10.1007/s10711-010-9469-8 (Original work published 2010)