TY - UNPB
T1 - Uniform boundedness for algebraic groups and Lie groups
AU - Kędra, Jarek
AU - Libman, Assaf
AU - Martin, Ben
N1 - Preliminary version. 11 pages
PY - 2022/2/28
Y1 - 2022/2/28
N2 - Let $G$ be a semisimple linear algebraic group over a field $k$ and let $G^+(k)$ be the subgroup generated by the subgroups $R_u(Q)(k)$, where $Q$ ranges over all the minimal $k$-parabolic subgroups $Q$ of $G$. We prove that if $G^+(k)$ is bounded then it is uniformly bounded. Under extra assumptions we get explicit bounds for $\Delta(G^+(k))$: we prove that if $k$ is algebraically closed then $\Delta(G^+(k))\leq 4\, {\rm rank}\,G$, and if $G$ is split over $k$ then $\Delta(G^+(k))\leq 28\, {\rm rank}\,G$. We deduce some analogous results for real and complex semisimple Lie groups.
AB - Let $G$ be a semisimple linear algebraic group over a field $k$ and let $G^+(k)$ be the subgroup generated by the subgroups $R_u(Q)(k)$, where $Q$ ranges over all the minimal $k$-parabolic subgroups $Q$ of $G$. We prove that if $G^+(k)$ is bounded then it is uniformly bounded. Under extra assumptions we get explicit bounds for $\Delta(G^+(k))$: we prove that if $k$ is algebraically closed then $\Delta(G^+(k))\leq 4\, {\rm rank}\,G$, and if $G$ is split over $k$ then $\Delta(G^+(k))\leq 28\, {\rm rank}\,G$. We deduce some analogous results for real and complex semisimple Lie groups.
KW - math.GR
KW - 20G15 (Primary) 20B07, 22E15 (Secondary)
UR - http://arxiv.org/abs/2202.13885v1
U2 - 10.48550/arXiv.2202.13885
DO - 10.48550/arXiv.2202.13885
M3 - Preprint
BT - Uniform boundedness for algebraic groups and Lie groups
PB - ArXiv
ER -