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Limits Under Conjugacy of the Diagonal Subgroup in SL(n,R)

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 Added by Arielle Leitner
 Publication date 2014
  fields
and research's language is English




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We give a quadratic lower bound on the dimension of the space of conjugacy classes of subgroups of SL(n,R) that are limits under conjugacy of the diagonal subgroup. We give the first explicit examples of abelian n-1 dimensional subgroups of SL(n,R) which are not such a limit, however all such abelian groups are limits of the diagonal group iff n < 5.

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139 - Arielle Leitner 2014
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We study the Chabauty compactification of two families of closed subgroups of $SL(n,mathbb{Q}_p)$. The first family is the set of all parahoric subgroups of $SL(n,mathbb{Q}_p)$. Although the Chabauty compactification of parahoric subgroups is well studied, we give a different and more geometric proof using various Levi decompositions of $SL(n,mathbb{Q}_p)$. Let $C$ be the subgroup of diagonal matrices in $SL(n, mathbb{Q}_p)$. The second family is the set of all $SL(n,mathbb{Q}_p)$-conjugates of $C$. We give a classification of the Chabauty limits of conjugates of $C$ using the action of $SL(n,mathbb{Q}_p)$ on its associated Bruhat--Tits building and compute all of the limits for $nleq 4$ (up to conjugacy). In contrast, for $ngeq 7$ we prove there are infinitely many $SL(n,mathbb{Q}_p)$-nonconjugate Chabauty limits of conjugates of $C$. Along the way we construct an explicit homeomorphism between the Chabauty compactification in $mathfrak{sl}(n, mathbb{Q}_p)$ of $SL(n,mathbb{Q}_p)$-conjugates of the $p$-adic Lie algebra of $C$ and the Chabauty compactification of $SL(n,mathbb{Q}_p)$-conjugates of $C$.
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