Rapinchuk's bounded-generation conjecture for higher-rank arithmetic groups

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Let Γ\Gamma be a noncocompact irreducible arithmetic group, and let rank⁡RΓ\operatorname{rank}_{\mathbb{R}}\Gamma denote its real rank. A group is boundedly generated by unipotent subgroups if it is a product of finitely many unipotent subgroups. Rapinchuk's conjecture. If rank⁡RΓ>1\operatorname{rank}_{\mathbb{R}}\Gamma>1, then Γ\Gamma contains a finite-index subgroup that is boundedly generated by unipotent subgroups. This conjecture is presented as a fundamental arithmetic-group statement that would imply the preceding noncocompact SL⁡(3)\operatorname{SL}(3) action conjecture through a theorem on actions on the line. Its resolution is not stated.

References

Primary source

Dave Witte Morris, “Some arithmetic groups that do not act on the circle”, arXiv:1210.3671 (2012).

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