Geodesic
The Length and Depth of Real Algebraic Groups
Journal of Lie theory, Tome 30 (2020) no. 3, pp. 627-652
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Let $G$ be a connected real algebraic group. An unrefinable chain of $G$ is a chain of subgroups $G=G_0>G_1>...>G_t=1$ where each $G_i$ is a maximal connected real subgroup of $G_{i-1}$. The maximal (respectively, minimal) length of such an unrefinable chain is called the length (respectively, depth) of $G$. We give a precise formula for the length of $G$, which generalises results of Burness, Liebeck and Shalev on complex algebraic groups and also on compact Lie groups. If $G$ is simple then we bound the depth of $G$ above and below, and in many cases we compute the exact value. In particular, the depth of any simple $G$ is at most $9$.
Classification : 20G20
Mots-clés : Length, depth, real algebraic groups 
@article{JLT_2020_30_3_JLT_2020_30_3_a1,
     author = {D. Sercombe},
     title = {The {Length} and {Depth} of {Real} {Algebraic} {Groups}},
     journal = {Journal of Lie theory},
     pages = {627--652},
     year = {2020},
     volume = {30},
     number = {3},
     url = {http://geodesic.mathdoc.fr/item/JLT_2020_30_3_JLT_2020_30_3_a1/}
}
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D. Sercombe. The Length and Depth of Real Algebraic Groups. Journal of Lie theory, Tome 30 (2020) no. 3, pp. 627-652. http://geodesic.mathdoc.fr/item/JLT_2020_30_3_JLT_2020_30_3_a1/