User davis - MathOverflow

Decomposing maximal compact subgroups of SO(n,1)

2013-02-22T02:48:28Z

Let $G=SO(n,1)$ and let $G=KAN$ be an Iwasawa decomposition of $G$. Let $M$ be the centralizer of $A$ in $K$. In this case, we have $K≃SO(n)$, $A≃\Bbb R$(this is the maximal diagonalizable subgroup), $N≃\Bbb{R}^{n−1}$ and $M≃SO(n−1)$. Let $\mathfrak k$ be the Lie algebra of $K$ and $\mathfrak m\subseteq\mathfrak k$ be the lie algebra of $M$. Let $\mathfrak h$ be the orthocomplement to $\mathfrak m$ in $\mathfrak k$. That is $\mathfrak k= \mathfrak m\oplus \mathfrak h$. Let $H$ be the Lie hgroup associated to generate by $exp(\mathfrak h)$. What can we say about $HM$? When is it true that $HM=K$?

I believe $K/HM$ is discrete and finite. In this case there exists $e=\omega_1,\omega_2,\dots,\omega_l$ s.t. $K=HM\sqcup\omega_2 HM\sqcup\dots\sqcup\omega_l HM$. Can I explicitly calculate these $\omega_i$?

Nilpotent subgroups of uniform finite index

2013-03-16T20:17:10Z

Let $G$ be a Lie group, $K\subseteq G$ be a compact group and $N\subseteq$ be a nilpotent group s.t. $N\cap K= {e}$. Let $H=N\rtimes K$ be the semidirect product of $N$ and $K$ and let $\Gamma$ be a discrete subgroup of $H$. Is it true that $\Gamma$ has a nilpotent subgroup of finite index. Also, can we guaranty that this index is uniform over all discrete $\Gamma\in H$. In other words, can we find a constant $C$ s.t. for all discrete subgroups $\Gamma$ of $H$, there exists $\tilde{\Gamma}$ a subgroup of $\Gamma$ s.t. $[\Gamma:\tilde{\Gamma}]\leq C$.

Maximal nilpotent subgroups of SO(n,1)

2013-02-09T04:52:54Z

For the Lie group $SO(n,1)$ I believe the maximal nilpotent subgroups are conjugate to either a diagonal group times a compact group or a unipotent group times a compact group. In either case the compact group will commute with the other group. Is this true and if so how do I prove it?

Extending a discrete sub group to a lattice in unimodular Lie groups

2013-01-31T15:08:07Z

Given a unimodular Lie group $G$ and a discrete subgroup $\Gamma\subseteq G$, under what conditions does there exists a discrete subgroup $H$ s.t. $\Gamma\subseteq H$ and $G/H$ has finite volume? Also, can someone give an example of a discrete subgroup $\Gamma$ of a unimodular Lie Group which cannot be extended to a lattice?

Comment by Davis

2013-03-17T16:05:00Z

Do you recall where in Raghunathan's book these results are? I find that book particularly hard to navigate through.

Comment by Davis

2013-02-26T04:11:14Z

Would it make more sense to consider quotients of S^n by discrete subgroups of isom $S^n$. My question is motivated by the fact that we have a natural understanding of what quotients of $E^n$ by discrete sub groups and Quotients of $H^n$ by discrete groups of the group of isometries are extensively studied. I am curious to know what kind of work if any is done on $S^n$.

Comment by Davis

2013-02-22T05:35:15Z

@Misha. Thank you. Indeed, $\mathfrak h$ is not a Lie Algebra. For my purpose it is enough to consider the group generated by $exp(\mathfrak h)$. In this case, what can I say about the cosets of $K/HM$?

Comment by Davis

2013-02-10T03:14:16Z

@Yves I do mean maximal connected nilpotent. Thank you.

Comment by Davis

2013-01-31T17:20:29Z

A quick follow up. When you re-embed $\Gamma$ into another Lie group, does the original group $G$ also embed into that group?