“p-adic” presentation of surfaces
On several occasions I heard about the following result:
For "certain" lattices $Lambda$ in $SL_2(mathbb{R})$, and almost any prime $p$ there exists a lattice $Gamma$ in $SL_2(mathbb{R})times SL_2(mathbb{Q}_p)$ and a compact subgroup $K$ of $SL_2(mathbb{R})times SL_2(mathbb{Q}_p)$ such that there is an isomorphism between
$$
Lambda backslash SL_2(mathbb{R})
$$
and
$$
Gamma backslash SL_2(mathbb{R})times SL_2(mathbb{Q}_p)/K.
$$
I know how to prove this for $Lambda = SL_2(mathbb{Z})$. Then $Gamma = SL_2(mathbb{Z}[1/p])$ (diagonally in $SL_2(mathbb{R})times SL_2(mathbb{Q}_p)$) and $K={1}times SL_2(mathbb{Z}_p)$ and the isomorphism is a quite easy map.
I would like to find a reference for more general $Lambda$, preferably with an explicit statement of the isomorphism and an explanation, what means "certain". Any help is highly appreciated!
group-theory reference-request hyperbolic-geometry lattices-in-lie-groups
edited Dec 3 '18 at 1:29
Paul Plummer
5,20721950
asked Jan 13 '13 at 10:22
Roger Weilik
211
add a comment |
On several occasions I heard about the following result:
For "certain" lattices $Lambda$ in $SL_2(mathbb{R})$, and almost any prime $p$ there exists a lattice $Gamma$ in $SL_2(mathbb{R})times SL_2(mathbb{Q}_p)$ and a compact subgroup $K$ of $SL_2(mathbb{R})times SL_2(mathbb{Q}_p)$ such that there is an isomorphism between
$$
Lambda backslash SL_2(mathbb{R})
$$
and
$$
Gamma backslash SL_2(mathbb{R})times SL_2(mathbb{Q}_p)/K.
$$
I know how to prove this for $Lambda = SL_2(mathbb{Z})$. Then $Gamma = SL_2(mathbb{Z}[1/p])$ (diagonally in $SL_2(mathbb{R})times SL_2(mathbb{Q}_p)$) and $K={1}times SL_2(mathbb{Z}_p)$ and the isomorphism is a quite easy map.
I would like to find a reference for more general $Lambda$, preferably with an explicit statement of the isomorphism and an explanation, what means "certain". Any help is highly appreciated!
group-theory reference-request hyperbolic-geometry lattices-in-lie-groups
edited Dec 3 '18 at 1:29
Paul Plummer
5,20721950
asked Jan 13 '13 at 10:22
Roger Weilik
211
Could you explain what you mean by "isomorphism"? (could you also put parentheses in your double coset? it took me 1 minute to read it: $Gammabackslash (dotstimesdots)/K$).
– YCor
Jan 18 '13 at 23:49
add a comment |
On several occasions I heard about the following result:
For "certain" lattices $Lambda$ in $SL_2(mathbb{R})$, and almost any prime $p$ there exists a lattice $Gamma$ in $SL_2(mathbb{R})times SL_2(mathbb{Q}_p)$ and a compact subgroup $K$ of $SL_2(mathbb{R})times SL_2(mathbb{Q}_p)$ such that there is an isomorphism between
$$
Lambda backslash SL_2(mathbb{R})
$$
and
$$
Gamma backslash SL_2(mathbb{R})times SL_2(mathbb{Q}_p)/K.
$$
I know how to prove this for $Lambda = SL_2(mathbb{Z})$. Then $Gamma = SL_2(mathbb{Z}[1/p])$ (diagonally in $SL_2(mathbb{R})times SL_2(mathbb{Q}_p)$) and $K={1}times SL_2(mathbb{Z}_p)$ and the isomorphism is a quite easy map.
I would like to find a reference for more general $Lambda$, preferably with an explicit statement of the isomorphism and an explanation, what means "certain". Any help is highly appreciated!
group-theory reference-request hyperbolic-geometry lattices-in-lie-groups
edited Dec 3 '18 at 1:29
Paul Plummer
5,20721950
asked Jan 13 '13 at 10:22
Roger Weilik
211
On several occasions I heard about the following result:
For "certain" lattices $Lambda$ in $SL_2(mathbb{R})$, and almost any prime $p$ there exists a lattice $Gamma$ in $SL_2(mathbb{R})times SL_2(mathbb{Q}_p)$ and a compact subgroup $K$ of $SL_2(mathbb{R})times SL_2(mathbb{Q}_p)$ such that there is an isomorphism between
$$
Lambda backslash SL_2(mathbb{R})
$$
and
$$
Gamma backslash SL_2(mathbb{R})times SL_2(mathbb{Q}_p)/K.
$$
I know how to prove this for $Lambda = SL_2(mathbb{Z})$. Then $Gamma = SL_2(mathbb{Z}[1/p])$ (diagonally in $SL_2(mathbb{R})times SL_2(mathbb{Q}_p)$) and $K={1}times SL_2(mathbb{Z}_p)$ and the isomorphism is a quite easy map.
I would like to find a reference for more general $Lambda$, preferably with an explicit statement of the isomorphism and an explanation, what means "certain". Any help is highly appreciated!
group-theory reference-request hyperbolic-geometry lattices-in-lie-groups
group-theory reference-request hyperbolic-geometry lattices-in-lie-groups
edited Dec 3 '18 at 1:29
Paul Plummer
5,20721950
asked Jan 13 '13 at 10:22
Roger Weilik
211
edited Dec 3 '18 at 1:29
Paul Plummer
5,20721950
asked Jan 13 '13 at 10:22
Roger Weilik
211
edited Dec 3 '18 at 1:29
Paul Plummer
5,20721950
edited Dec 3 '18 at 1:29
Paul Plummer
5,20721950
edited Dec 3 '18 at 1:29
Paul Plummer
5,20721950
5,20721950
asked Jan 13 '13 at 10:22
Roger Weilik
211
asked Jan 13 '13 at 10:22
Roger Weilik
211
asked Jan 13 '13 at 10:22
Roger Weilik
211
211
Could you explain what you mean by "isomorphism"? (could you also put parentheses in your double coset? it took me 1 minute to read it: $Gammabackslash (dotstimesdots)/K$).
– YCor
Jan 18 '13 at 23:49
add a comment |
Could you explain what you mean by "isomorphism"? (could you also put parentheses in your double coset? it took me 1 minute to read it: $Gammabackslash (dotstimesdots)/K$).
– YCor
Jan 18 '13 at 23:49
Could you explain what you mean by "isomorphism"? (could you also put parentheses in your double coset? it took me 1 minute to read it: $Gammabackslash (dotstimesdots)/K$).
– YCor
Jan 18 '13 at 23:49
Could you explain what you mean by "isomorphism"? (could you also put parentheses in your double coset? it took me 1 minute to read it: $Gammabackslash (dotstimesdots)/K$).
– YCor
Jan 18 '13 at 23:49
add a comment |
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Could you explain what you mean by "isomorphism"? (could you also put parentheses in your double coset? it took me 1 minute to read it: $Gammabackslash (dotstimesdots)/K$).
– YCor
Jan 18 '13 at 23:49