Virtual simplicity conjecture for lattices in exotic affine buildings

Let XX be an affine building of type A~2\widetilde A_2 that is not isomorphic to the building associated to PGL3(D)\mathrm{PGL}_3(D), where DD is a finite-dimensional division algebra over a local field, and let Γ\Gamma be a lattice in Aut(X)\operatorname{Aut}(X). Virtual simplicity conjecture. Then Γ\Gamma is virtually simple. The question concerns normal subgroups of lattices in possibly exotic affine buildings; finite-index normal subgroups are abundant for linear lattices, while the existence of finite quotients for non-linear lattices is not understood.

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Primary source

Uri Bader, Pierre-Emmanuel Caprace and Jean Lécureux, “On the linearity of lattices in affine buildings and ergodicity of the singular Cartan flow”, arXiv:1608.06265 (2018).

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