“p-adic” presentation of surfaces
$begingroup$
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
$endgroup$
add a comment |
$begingroup$
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
$endgroup$
$begingroup$
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: $Gammabackslash (dotstimesdots)/K$).
$endgroup$
– YCor
Jan 18 '13 at 23:49
add a comment |
$begingroup$
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
$endgroup$
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
5,20721950
asked Jan 13 '13 at 10:22
Roger WeilikRoger Weilik
211
211
$begingroup$
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: $Gammabackslash (dotstimesdots)/K$).
$endgroup$
– YCor
Jan 18 '13 at 23:49
add a comment |
$begingroup$
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: $Gammabackslash (dotstimesdots)/K$).
$endgroup$
– YCor
Jan 18 '13 at 23:49
$begingroup$
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: $Gammabackslash (dotstimesdots)/K$).
$endgroup$
– YCor
Jan 18 '13 at 23:49
$begingroup$
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: $Gammabackslash (dotstimesdots)/K$).
$endgroup$
– YCor
Jan 18 '13 at 23:49
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f276850%2fp-adic-presentation-of-surfaces%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f276850%2fp-adic-presentation-of-surfaces%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
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: $Gammabackslash (dotstimesdots)/K$).
$endgroup$
– YCor
Jan 18 '13 at 23:49