Do general discrete subgroups of $operatorname{SL}_2(mathbb R)$ have fundamental domains in the upper half...












2












$begingroup$


Let $Gamma$ be a congruence subgroup of $operatorname{SL}_2(mathbb Z)$. The quotient $Gamma backslash mathbb H$ has the structure of a one dimensional complex manifold, such that the quotient map $pi: mathbb H rightarrow Gamma backslash mathbb H$ is holomorphic. There is a nice fundamental domain $D subset mathbb H$ coming from the usual fundamental domain for $operatorname{SL}_2(mathbb Z)$ which, up to some boundary identification, gives us a copy of $Gamma backslash mathbb H$ inside $mathbb H$.



The Borel measure $mu = frac{dx dy}{y^2}$ on $mathbb H$ descends to a Borel measure $overline{mu}$ on $Gamma backslash mathbb H$ which we may define using the fundamental domain: if $U subset Gamma backslash mathbb H$ is Borel, then we set



$$overline{mu}(U) := mu bigg(pi^{-1}(U) cap D bigg) tag{1}$$



Now, assume $Gamma$ is an arbitrary discrete subgroup of $operatorname{SL}_2(mathbb R)$.





  1. Is there a canonical measure $bar{mu}$ on $Gamma backslash mathbb H$ coming from $mu = frac{dx dy}{y^2}$?


  2. Does there always exist a fundamental domain $D$ for $Gamma$?


  3. Can $overline{mu}$ arise from a differential form on $Gamma backslash mathbb H$? That is, does $overline{mu}$ come from a (unique?) smooth differential $2$-form $overline{omega}$ on $Gamma backslash mathbb H$ (thought of as a smooth manifold) which pulls back to the differential form on the smooth manifold $mathbb H$ corresponding to $mu = frac{dx dy}{y^2}$?





For intuition, I'm thinking of $mathbb R$ modulo the action of $mathbb Z$. Up to boundary identification, $[0,1]$ is a fundamental domain for the action of $mathbb Z$ on $mathbb R$. For the Haar measure $bar{mu}$ on $mathbb R/mathbb Z$, we can get it in two ways. First, if $pi: mathbb R rightarrow mathbb Z$ is the quotient map, we can measure subsets of $mathbb R/mathbb Z$ by pulling them back to $mathbb R$, intersecting them with $[0,1]$, then measuring. Second, $bar{mu}$ comes from the unique invariant nonvanishing $1$-form on $mathbb R/mathbb Z$ which pulls back to the top form $dx$ on $mathbb R$ giving the Lebesgue measure on $mathbb R$.










share|cite|improve this question









$endgroup$












  • $begingroup$
    If $Gamma$ is a discrete subgroup of $G$, won't the way $K$ acts on the $3$-dimensional locally isomorphic real manifolds $G$ and $Gamma setminus G$ ($1$-dimensional continuous and compact right action of $K$) means that $Gamma setminus G/ K$ is a $2$-dimensional real manifold whose topology is locally isomorphic to that of $G/Kcong mathbb{H}$, this gives the fundamental domain, and considering $G/K$ as a $1$-dimensional complex manifold (a Riemann surface) then so is $Gamma setminus G/ K$ ?
    $endgroup$
    – reuns
    Jan 7 at 2:38












  • $begingroup$
    Where is the fundamental domain? I don't understand
    $endgroup$
    – D_S
    Jan 7 at 2:45










  • $begingroup$
    I'm not immediately convinced..there is also the issue that generally $G/K rightarrow Gamma backslash G/K$ is not a local homeomorphism quite everywhere because you might have some elliptic points.
    $endgroup$
    – D_S
    Jan 7 at 3:00










  • $begingroup$
    Elliptic points (that is $gamma g K = g K, gamma not in g Kg^{-1}$) are a good point and quite the main issue then. And it seems they are automatically removed from the Dirichlet domain. Anyway starting with the $G$ invariant metric on $G/K$ should make things easier.
    $endgroup$
    – reuns
    Jan 7 at 4:40


















2












$begingroup$


Let $Gamma$ be a congruence subgroup of $operatorname{SL}_2(mathbb Z)$. The quotient $Gamma backslash mathbb H$ has the structure of a one dimensional complex manifold, such that the quotient map $pi: mathbb H rightarrow Gamma backslash mathbb H$ is holomorphic. There is a nice fundamental domain $D subset mathbb H$ coming from the usual fundamental domain for $operatorname{SL}_2(mathbb Z)$ which, up to some boundary identification, gives us a copy of $Gamma backslash mathbb H$ inside $mathbb H$.



The Borel measure $mu = frac{dx dy}{y^2}$ on $mathbb H$ descends to a Borel measure $overline{mu}$ on $Gamma backslash mathbb H$ which we may define using the fundamental domain: if $U subset Gamma backslash mathbb H$ is Borel, then we set



$$overline{mu}(U) := mu bigg(pi^{-1}(U) cap D bigg) tag{1}$$



Now, assume $Gamma$ is an arbitrary discrete subgroup of $operatorname{SL}_2(mathbb R)$.





  1. Is there a canonical measure $bar{mu}$ on $Gamma backslash mathbb H$ coming from $mu = frac{dx dy}{y^2}$?


  2. Does there always exist a fundamental domain $D$ for $Gamma$?


  3. Can $overline{mu}$ arise from a differential form on $Gamma backslash mathbb H$? That is, does $overline{mu}$ come from a (unique?) smooth differential $2$-form $overline{omega}$ on $Gamma backslash mathbb H$ (thought of as a smooth manifold) which pulls back to the differential form on the smooth manifold $mathbb H$ corresponding to $mu = frac{dx dy}{y^2}$?





For intuition, I'm thinking of $mathbb R$ modulo the action of $mathbb Z$. Up to boundary identification, $[0,1]$ is a fundamental domain for the action of $mathbb Z$ on $mathbb R$. For the Haar measure $bar{mu}$ on $mathbb R/mathbb Z$, we can get it in two ways. First, if $pi: mathbb R rightarrow mathbb Z$ is the quotient map, we can measure subsets of $mathbb R/mathbb Z$ by pulling them back to $mathbb R$, intersecting them with $[0,1]$, then measuring. Second, $bar{mu}$ comes from the unique invariant nonvanishing $1$-form on $mathbb R/mathbb Z$ which pulls back to the top form $dx$ on $mathbb R$ giving the Lebesgue measure on $mathbb R$.










share|cite|improve this question









$endgroup$












  • $begingroup$
    If $Gamma$ is a discrete subgroup of $G$, won't the way $K$ acts on the $3$-dimensional locally isomorphic real manifolds $G$ and $Gamma setminus G$ ($1$-dimensional continuous and compact right action of $K$) means that $Gamma setminus G/ K$ is a $2$-dimensional real manifold whose topology is locally isomorphic to that of $G/Kcong mathbb{H}$, this gives the fundamental domain, and considering $G/K$ as a $1$-dimensional complex manifold (a Riemann surface) then so is $Gamma setminus G/ K$ ?
    $endgroup$
    – reuns
    Jan 7 at 2:38












  • $begingroup$
    Where is the fundamental domain? I don't understand
    $endgroup$
    – D_S
    Jan 7 at 2:45










  • $begingroup$
    I'm not immediately convinced..there is also the issue that generally $G/K rightarrow Gamma backslash G/K$ is not a local homeomorphism quite everywhere because you might have some elliptic points.
    $endgroup$
    – D_S
    Jan 7 at 3:00










  • $begingroup$
    Elliptic points (that is $gamma g K = g K, gamma not in g Kg^{-1}$) are a good point and quite the main issue then. And it seems they are automatically removed from the Dirichlet domain. Anyway starting with the $G$ invariant metric on $G/K$ should make things easier.
    $endgroup$
    – reuns
    Jan 7 at 4:40
















2












2








2


1



$begingroup$


Let $Gamma$ be a congruence subgroup of $operatorname{SL}_2(mathbb Z)$. The quotient $Gamma backslash mathbb H$ has the structure of a one dimensional complex manifold, such that the quotient map $pi: mathbb H rightarrow Gamma backslash mathbb H$ is holomorphic. There is a nice fundamental domain $D subset mathbb H$ coming from the usual fundamental domain for $operatorname{SL}_2(mathbb Z)$ which, up to some boundary identification, gives us a copy of $Gamma backslash mathbb H$ inside $mathbb H$.



The Borel measure $mu = frac{dx dy}{y^2}$ on $mathbb H$ descends to a Borel measure $overline{mu}$ on $Gamma backslash mathbb H$ which we may define using the fundamental domain: if $U subset Gamma backslash mathbb H$ is Borel, then we set



$$overline{mu}(U) := mu bigg(pi^{-1}(U) cap D bigg) tag{1}$$



Now, assume $Gamma$ is an arbitrary discrete subgroup of $operatorname{SL}_2(mathbb R)$.





  1. Is there a canonical measure $bar{mu}$ on $Gamma backslash mathbb H$ coming from $mu = frac{dx dy}{y^2}$?


  2. Does there always exist a fundamental domain $D$ for $Gamma$?


  3. Can $overline{mu}$ arise from a differential form on $Gamma backslash mathbb H$? That is, does $overline{mu}$ come from a (unique?) smooth differential $2$-form $overline{omega}$ on $Gamma backslash mathbb H$ (thought of as a smooth manifold) which pulls back to the differential form on the smooth manifold $mathbb H$ corresponding to $mu = frac{dx dy}{y^2}$?





For intuition, I'm thinking of $mathbb R$ modulo the action of $mathbb Z$. Up to boundary identification, $[0,1]$ is a fundamental domain for the action of $mathbb Z$ on $mathbb R$. For the Haar measure $bar{mu}$ on $mathbb R/mathbb Z$, we can get it in two ways. First, if $pi: mathbb R rightarrow mathbb Z$ is the quotient map, we can measure subsets of $mathbb R/mathbb Z$ by pulling them back to $mathbb R$, intersecting them with $[0,1]$, then measuring. Second, $bar{mu}$ comes from the unique invariant nonvanishing $1$-form on $mathbb R/mathbb Z$ which pulls back to the top form $dx$ on $mathbb R$ giving the Lebesgue measure on $mathbb R$.










share|cite|improve this question









$endgroup$




Let $Gamma$ be a congruence subgroup of $operatorname{SL}_2(mathbb Z)$. The quotient $Gamma backslash mathbb H$ has the structure of a one dimensional complex manifold, such that the quotient map $pi: mathbb H rightarrow Gamma backslash mathbb H$ is holomorphic. There is a nice fundamental domain $D subset mathbb H$ coming from the usual fundamental domain for $operatorname{SL}_2(mathbb Z)$ which, up to some boundary identification, gives us a copy of $Gamma backslash mathbb H$ inside $mathbb H$.



The Borel measure $mu = frac{dx dy}{y^2}$ on $mathbb H$ descends to a Borel measure $overline{mu}$ on $Gamma backslash mathbb H$ which we may define using the fundamental domain: if $U subset Gamma backslash mathbb H$ is Borel, then we set



$$overline{mu}(U) := mu bigg(pi^{-1}(U) cap D bigg) tag{1}$$



Now, assume $Gamma$ is an arbitrary discrete subgroup of $operatorname{SL}_2(mathbb R)$.





  1. Is there a canonical measure $bar{mu}$ on $Gamma backslash mathbb H$ coming from $mu = frac{dx dy}{y^2}$?


  2. Does there always exist a fundamental domain $D$ for $Gamma$?


  3. Can $overline{mu}$ arise from a differential form on $Gamma backslash mathbb H$? That is, does $overline{mu}$ come from a (unique?) smooth differential $2$-form $overline{omega}$ on $Gamma backslash mathbb H$ (thought of as a smooth manifold) which pulls back to the differential form on the smooth manifold $mathbb H$ corresponding to $mu = frac{dx dy}{y^2}$?





For intuition, I'm thinking of $mathbb R$ modulo the action of $mathbb Z$. Up to boundary identification, $[0,1]$ is a fundamental domain for the action of $mathbb Z$ on $mathbb R$. For the Haar measure $bar{mu}$ on $mathbb R/mathbb Z$, we can get it in two ways. First, if $pi: mathbb R rightarrow mathbb Z$ is the quotient map, we can measure subsets of $mathbb R/mathbb Z$ by pulling them back to $mathbb R$, intersecting them with $[0,1]$, then measuring. Second, $bar{mu}$ comes from the unique invariant nonvanishing $1$-form on $mathbb R/mathbb Z$ which pulls back to the top form $dx$ on $mathbb R$ giving the Lebesgue measure on $mathbb R$.







complex-analysis number-theory measure-theory differential-geometry modular-forms






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share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jan 7 at 0:00









D_SD_S

14.2k61754




14.2k61754












  • $begingroup$
    If $Gamma$ is a discrete subgroup of $G$, won't the way $K$ acts on the $3$-dimensional locally isomorphic real manifolds $G$ and $Gamma setminus G$ ($1$-dimensional continuous and compact right action of $K$) means that $Gamma setminus G/ K$ is a $2$-dimensional real manifold whose topology is locally isomorphic to that of $G/Kcong mathbb{H}$, this gives the fundamental domain, and considering $G/K$ as a $1$-dimensional complex manifold (a Riemann surface) then so is $Gamma setminus G/ K$ ?
    $endgroup$
    – reuns
    Jan 7 at 2:38












  • $begingroup$
    Where is the fundamental domain? I don't understand
    $endgroup$
    – D_S
    Jan 7 at 2:45










  • $begingroup$
    I'm not immediately convinced..there is also the issue that generally $G/K rightarrow Gamma backslash G/K$ is not a local homeomorphism quite everywhere because you might have some elliptic points.
    $endgroup$
    – D_S
    Jan 7 at 3:00










  • $begingroup$
    Elliptic points (that is $gamma g K = g K, gamma not in g Kg^{-1}$) are a good point and quite the main issue then. And it seems they are automatically removed from the Dirichlet domain. Anyway starting with the $G$ invariant metric on $G/K$ should make things easier.
    $endgroup$
    – reuns
    Jan 7 at 4:40




















  • $begingroup$
    If $Gamma$ is a discrete subgroup of $G$, won't the way $K$ acts on the $3$-dimensional locally isomorphic real manifolds $G$ and $Gamma setminus G$ ($1$-dimensional continuous and compact right action of $K$) means that $Gamma setminus G/ K$ is a $2$-dimensional real manifold whose topology is locally isomorphic to that of $G/Kcong mathbb{H}$, this gives the fundamental domain, and considering $G/K$ as a $1$-dimensional complex manifold (a Riemann surface) then so is $Gamma setminus G/ K$ ?
    $endgroup$
    – reuns
    Jan 7 at 2:38












  • $begingroup$
    Where is the fundamental domain? I don't understand
    $endgroup$
    – D_S
    Jan 7 at 2:45










  • $begingroup$
    I'm not immediately convinced..there is also the issue that generally $G/K rightarrow Gamma backslash G/K$ is not a local homeomorphism quite everywhere because you might have some elliptic points.
    $endgroup$
    – D_S
    Jan 7 at 3:00










  • $begingroup$
    Elliptic points (that is $gamma g K = g K, gamma not in g Kg^{-1}$) are a good point and quite the main issue then. And it seems they are automatically removed from the Dirichlet domain. Anyway starting with the $G$ invariant metric on $G/K$ should make things easier.
    $endgroup$
    – reuns
    Jan 7 at 4:40


















$begingroup$
If $Gamma$ is a discrete subgroup of $G$, won't the way $K$ acts on the $3$-dimensional locally isomorphic real manifolds $G$ and $Gamma setminus G$ ($1$-dimensional continuous and compact right action of $K$) means that $Gamma setminus G/ K$ is a $2$-dimensional real manifold whose topology is locally isomorphic to that of $G/Kcong mathbb{H}$, this gives the fundamental domain, and considering $G/K$ as a $1$-dimensional complex manifold (a Riemann surface) then so is $Gamma setminus G/ K$ ?
$endgroup$
– reuns
Jan 7 at 2:38






$begingroup$
If $Gamma$ is a discrete subgroup of $G$, won't the way $K$ acts on the $3$-dimensional locally isomorphic real manifolds $G$ and $Gamma setminus G$ ($1$-dimensional continuous and compact right action of $K$) means that $Gamma setminus G/ K$ is a $2$-dimensional real manifold whose topology is locally isomorphic to that of $G/Kcong mathbb{H}$, this gives the fundamental domain, and considering $G/K$ as a $1$-dimensional complex manifold (a Riemann surface) then so is $Gamma setminus G/ K$ ?
$endgroup$
– reuns
Jan 7 at 2:38














$begingroup$
Where is the fundamental domain? I don't understand
$endgroup$
– D_S
Jan 7 at 2:45




$begingroup$
Where is the fundamental domain? I don't understand
$endgroup$
– D_S
Jan 7 at 2:45












$begingroup$
I'm not immediately convinced..there is also the issue that generally $G/K rightarrow Gamma backslash G/K$ is not a local homeomorphism quite everywhere because you might have some elliptic points.
$endgroup$
– D_S
Jan 7 at 3:00




$begingroup$
I'm not immediately convinced..there is also the issue that generally $G/K rightarrow Gamma backslash G/K$ is not a local homeomorphism quite everywhere because you might have some elliptic points.
$endgroup$
– D_S
Jan 7 at 3:00












$begingroup$
Elliptic points (that is $gamma g K = g K, gamma not in g Kg^{-1}$) are a good point and quite the main issue then. And it seems they are automatically removed from the Dirichlet domain. Anyway starting with the $G$ invariant metric on $G/K$ should make things easier.
$endgroup$
– reuns
Jan 7 at 4:40






$begingroup$
Elliptic points (that is $gamma g K = g K, gamma not in g Kg^{-1}$) are a good point and quite the main issue then. And it seems they are automatically removed from the Dirichlet domain. Anyway starting with the $G$ invariant metric on $G/K$ should make things easier.
$endgroup$
– reuns
Jan 7 at 4:40












3 Answers
3






active

oldest

votes


















4












$begingroup$

I'd recommend not thinking that choice of a "nice fundamental domain" is of much importance for the basic development of things. Yes, the details of a fundamental domain tell something about generators of the discrete group $Gamma$, but that's not universally necessary, nor intelligible.



So, for example, the key relationship between functions on $mathfrak H$ and $Gammabackslash mathfrak H$, or, equivalently, $Gammabackslash G$ and $G$, or $Gammabackslash G/K$, where $G=SL_2(mathbb R)$ and $K=SO(2,mathbb R)$, can be described very well without any mention or choice of "fundamental domain". This is fortunate. Namely, one proves the lemma that the averaging map $fto sum_{gammain Gamma} fcirc gamma$ from $C^o_c(G)$ to $C^o_c(Gammabackslash G)$ is surjective. Then the uniqueness of invariant distributions... shows that there is a unique integral/measure on $Gammabackslash G$ such that "unwinding" is correct, namely, that
$$
int_G f(g);dg ;=; int_{Gammabackslash G} sum_{gammainGamma} f(gamma g);ddot{g}
$$

with $ddot{g}$ denoting that measure on the quotient.



(Happily for us, the only things that this set-up depends upon are that $G$ be a unimodular topological group, and $Gamma$ a discrete subgroup.)






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Does the passage of measures from $G$ to $Gamma backslash G$ induce a passage of measures from $G/K$ to $Gamma backslash G /K$? That seems to be the remaining issue here
    $endgroup$
    – D_S
    Jan 7 at 0:45










  • $begingroup$
    @D_S, yes, the averaging map from $G$ to $Gammabackslash G$ commutes with the right $K$ action, and $K$ is compact, so we can likewise show that $K$-invariant things surject to $K$-invariant, etc. No problem.
    $endgroup$
    – paul garrett
    Jan 7 at 18:14










  • $begingroup$
    Thanks for your answer. I wrote an answer of my own explaining in my own words what you wrote. Would you mind seeing if there is anything incorrect about what I wrote? I want to make sure I understand your answer completely.
    $endgroup$
    – D_S
    Jan 8 at 3:49



















1












$begingroup$

To answer the question asked, yes, every discrete subgroup of $SL_2(mathbb R)$ has a fundamental domain, called a Dirichlet domain.






share|cite|improve this answer









$endgroup$





















    0












    $begingroup$

    Let $G = operatorname{SL}_2(mathbb R), K = operatorname{SO}_2(mathbb R)$, and $Gamma$ a discrete subgroup of $G$. We identify the measures $mu, dot mu, bar{mu}$ on $G, Gamma backslash G$ and $G/K$ respectively with positive linear functionals $T, dot T, bar{T}$ on $mathscr C_c(G), mathscr C_c(Gamma backslash G), mathscr C_c(G/K)$ respectively. For example,



    $$T(f) = intlimits_G f(g)dg tag{$f in mathscr C_c(G)$}$$



    $$bar{T}(f) = intlimits_{G/K} f(gK)dbar{mu}(gK) tag{$f in mathscr C_c(G/K)$}$$



    The upper half plane $mathbb H$ with the hyperbolic measure $frac{dxdy}{y^2}$ are identified with $G/K$ and $bar{mu}$, respectively.



    Since $K$ is compact, we have a natural identification of $mathscr C_c(G/K)$ with those elements of $mathscr C_c(G)$ which are right $K$-invariant, by precomposing with the quotient map $G rightarrow G/K$. We may similarly identify $mathscr C_c(Gammabackslash G/K)$ with the right $K$-invariant elements of $mathscr C_c(Gamma backslash G)$ via the quotient map $Gamma backslash G rightarrow Gamma backslash G/K$. The surjection



    $$delta: mathscr C_c(G) rightarrow mathscr C_c(Gamma backslash G)$$



    $$delta(f)(Gamma g) = sumlimits_{gamma in Gamma} f(gamma g)$$



    can be shown to restrict to a surjection



    $$mathscr C_c(G/K) rightarrow mathscr C_c(Gamma backslash G /K)$$



    The answer to my question follows if I can prove the following proposition.




    Proposition: There exists a unique linear positive functional $widetilde{T}$ on $mathscr C_c(Gamma backslash G/K)$, with corresponding measure $widetilde{mu}$, such that



    $$bar{T} = widetilde{T} circ delta$$




    In terms of measures, what this says is that if we define $widetilde{T}(f) =: intlimits_{Gamma backslash G/K} f(Gamma g K) dwidetilde{mu}(Gamma g K)$, then



    $$ intlimits_{G/K} f(gK) d bar{mu}(gK) = intlimits_{Gamma backslash G/K} space space spacebigg( sumlimits_{gamma in Gamma} space space f(gamma g K) bigg)space d widetilde{mu}(Gamma gK) $$



    or



    $$intlimits_{mathbb H} f(z)y^{-2} dxdy = intlimits_{Gamma backslash mathbb H} sumlimits_{gamma in Gamma} f(gamma.z) d widetilde{mu}(Gamma z)$$






    share|cite|improve this answer









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      3 Answers
      3






      active

      oldest

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      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      4












      $begingroup$

      I'd recommend not thinking that choice of a "nice fundamental domain" is of much importance for the basic development of things. Yes, the details of a fundamental domain tell something about generators of the discrete group $Gamma$, but that's not universally necessary, nor intelligible.



      So, for example, the key relationship between functions on $mathfrak H$ and $Gammabackslash mathfrak H$, or, equivalently, $Gammabackslash G$ and $G$, or $Gammabackslash G/K$, where $G=SL_2(mathbb R)$ and $K=SO(2,mathbb R)$, can be described very well without any mention or choice of "fundamental domain". This is fortunate. Namely, one proves the lemma that the averaging map $fto sum_{gammain Gamma} fcirc gamma$ from $C^o_c(G)$ to $C^o_c(Gammabackslash G)$ is surjective. Then the uniqueness of invariant distributions... shows that there is a unique integral/measure on $Gammabackslash G$ such that "unwinding" is correct, namely, that
      $$
      int_G f(g);dg ;=; int_{Gammabackslash G} sum_{gammainGamma} f(gamma g);ddot{g}
      $$

      with $ddot{g}$ denoting that measure on the quotient.



      (Happily for us, the only things that this set-up depends upon are that $G$ be a unimodular topological group, and $Gamma$ a discrete subgroup.)






      share|cite|improve this answer









      $endgroup$













      • $begingroup$
        Does the passage of measures from $G$ to $Gamma backslash G$ induce a passage of measures from $G/K$ to $Gamma backslash G /K$? That seems to be the remaining issue here
        $endgroup$
        – D_S
        Jan 7 at 0:45










      • $begingroup$
        @D_S, yes, the averaging map from $G$ to $Gammabackslash G$ commutes with the right $K$ action, and $K$ is compact, so we can likewise show that $K$-invariant things surject to $K$-invariant, etc. No problem.
        $endgroup$
        – paul garrett
        Jan 7 at 18:14










      • $begingroup$
        Thanks for your answer. I wrote an answer of my own explaining in my own words what you wrote. Would you mind seeing if there is anything incorrect about what I wrote? I want to make sure I understand your answer completely.
        $endgroup$
        – D_S
        Jan 8 at 3:49
















      4












      $begingroup$

      I'd recommend not thinking that choice of a "nice fundamental domain" is of much importance for the basic development of things. Yes, the details of a fundamental domain tell something about generators of the discrete group $Gamma$, but that's not universally necessary, nor intelligible.



      So, for example, the key relationship between functions on $mathfrak H$ and $Gammabackslash mathfrak H$, or, equivalently, $Gammabackslash G$ and $G$, or $Gammabackslash G/K$, where $G=SL_2(mathbb R)$ and $K=SO(2,mathbb R)$, can be described very well without any mention or choice of "fundamental domain". This is fortunate. Namely, one proves the lemma that the averaging map $fto sum_{gammain Gamma} fcirc gamma$ from $C^o_c(G)$ to $C^o_c(Gammabackslash G)$ is surjective. Then the uniqueness of invariant distributions... shows that there is a unique integral/measure on $Gammabackslash G$ such that "unwinding" is correct, namely, that
      $$
      int_G f(g);dg ;=; int_{Gammabackslash G} sum_{gammainGamma} f(gamma g);ddot{g}
      $$

      with $ddot{g}$ denoting that measure on the quotient.



      (Happily for us, the only things that this set-up depends upon are that $G$ be a unimodular topological group, and $Gamma$ a discrete subgroup.)






      share|cite|improve this answer









      $endgroup$













      • $begingroup$
        Does the passage of measures from $G$ to $Gamma backslash G$ induce a passage of measures from $G/K$ to $Gamma backslash G /K$? That seems to be the remaining issue here
        $endgroup$
        – D_S
        Jan 7 at 0:45










      • $begingroup$
        @D_S, yes, the averaging map from $G$ to $Gammabackslash G$ commutes with the right $K$ action, and $K$ is compact, so we can likewise show that $K$-invariant things surject to $K$-invariant, etc. No problem.
        $endgroup$
        – paul garrett
        Jan 7 at 18:14










      • $begingroup$
        Thanks for your answer. I wrote an answer of my own explaining in my own words what you wrote. Would you mind seeing if there is anything incorrect about what I wrote? I want to make sure I understand your answer completely.
        $endgroup$
        – D_S
        Jan 8 at 3:49














      4












      4








      4





      $begingroup$

      I'd recommend not thinking that choice of a "nice fundamental domain" is of much importance for the basic development of things. Yes, the details of a fundamental domain tell something about generators of the discrete group $Gamma$, but that's not universally necessary, nor intelligible.



      So, for example, the key relationship between functions on $mathfrak H$ and $Gammabackslash mathfrak H$, or, equivalently, $Gammabackslash G$ and $G$, or $Gammabackslash G/K$, where $G=SL_2(mathbb R)$ and $K=SO(2,mathbb R)$, can be described very well without any mention or choice of "fundamental domain". This is fortunate. Namely, one proves the lemma that the averaging map $fto sum_{gammain Gamma} fcirc gamma$ from $C^o_c(G)$ to $C^o_c(Gammabackslash G)$ is surjective. Then the uniqueness of invariant distributions... shows that there is a unique integral/measure on $Gammabackslash G$ such that "unwinding" is correct, namely, that
      $$
      int_G f(g);dg ;=; int_{Gammabackslash G} sum_{gammainGamma} f(gamma g);ddot{g}
      $$

      with $ddot{g}$ denoting that measure on the quotient.



      (Happily for us, the only things that this set-up depends upon are that $G$ be a unimodular topological group, and $Gamma$ a discrete subgroup.)






      share|cite|improve this answer









      $endgroup$



      I'd recommend not thinking that choice of a "nice fundamental domain" is of much importance for the basic development of things. Yes, the details of a fundamental domain tell something about generators of the discrete group $Gamma$, but that's not universally necessary, nor intelligible.



      So, for example, the key relationship between functions on $mathfrak H$ and $Gammabackslash mathfrak H$, or, equivalently, $Gammabackslash G$ and $G$, or $Gammabackslash G/K$, where $G=SL_2(mathbb R)$ and $K=SO(2,mathbb R)$, can be described very well without any mention or choice of "fundamental domain". This is fortunate. Namely, one proves the lemma that the averaging map $fto sum_{gammain Gamma} fcirc gamma$ from $C^o_c(G)$ to $C^o_c(Gammabackslash G)$ is surjective. Then the uniqueness of invariant distributions... shows that there is a unique integral/measure on $Gammabackslash G$ such that "unwinding" is correct, namely, that
      $$
      int_G f(g);dg ;=; int_{Gammabackslash G} sum_{gammainGamma} f(gamma g);ddot{g}
      $$

      with $ddot{g}$ denoting that measure on the quotient.



      (Happily for us, the only things that this set-up depends upon are that $G$ be a unimodular topological group, and $Gamma$ a discrete subgroup.)







      share|cite|improve this answer












      share|cite|improve this answer



      share|cite|improve this answer










      answered Jan 7 at 0:37









      paul garrettpaul garrett

      32.2k362120




      32.2k362120












      • $begingroup$
        Does the passage of measures from $G$ to $Gamma backslash G$ induce a passage of measures from $G/K$ to $Gamma backslash G /K$? That seems to be the remaining issue here
        $endgroup$
        – D_S
        Jan 7 at 0:45










      • $begingroup$
        @D_S, yes, the averaging map from $G$ to $Gammabackslash G$ commutes with the right $K$ action, and $K$ is compact, so we can likewise show that $K$-invariant things surject to $K$-invariant, etc. No problem.
        $endgroup$
        – paul garrett
        Jan 7 at 18:14










      • $begingroup$
        Thanks for your answer. I wrote an answer of my own explaining in my own words what you wrote. Would you mind seeing if there is anything incorrect about what I wrote? I want to make sure I understand your answer completely.
        $endgroup$
        – D_S
        Jan 8 at 3:49


















      • $begingroup$
        Does the passage of measures from $G$ to $Gamma backslash G$ induce a passage of measures from $G/K$ to $Gamma backslash G /K$? That seems to be the remaining issue here
        $endgroup$
        – D_S
        Jan 7 at 0:45










      • $begingroup$
        @D_S, yes, the averaging map from $G$ to $Gammabackslash G$ commutes with the right $K$ action, and $K$ is compact, so we can likewise show that $K$-invariant things surject to $K$-invariant, etc. No problem.
        $endgroup$
        – paul garrett
        Jan 7 at 18:14










      • $begingroup$
        Thanks for your answer. I wrote an answer of my own explaining in my own words what you wrote. Would you mind seeing if there is anything incorrect about what I wrote? I want to make sure I understand your answer completely.
        $endgroup$
        – D_S
        Jan 8 at 3:49
















      $begingroup$
      Does the passage of measures from $G$ to $Gamma backslash G$ induce a passage of measures from $G/K$ to $Gamma backslash G /K$? That seems to be the remaining issue here
      $endgroup$
      – D_S
      Jan 7 at 0:45




      $begingroup$
      Does the passage of measures from $G$ to $Gamma backslash G$ induce a passage of measures from $G/K$ to $Gamma backslash G /K$? That seems to be the remaining issue here
      $endgroup$
      – D_S
      Jan 7 at 0:45












      $begingroup$
      @D_S, yes, the averaging map from $G$ to $Gammabackslash G$ commutes with the right $K$ action, and $K$ is compact, so we can likewise show that $K$-invariant things surject to $K$-invariant, etc. No problem.
      $endgroup$
      – paul garrett
      Jan 7 at 18:14




      $begingroup$
      @D_S, yes, the averaging map from $G$ to $Gammabackslash G$ commutes with the right $K$ action, and $K$ is compact, so we can likewise show that $K$-invariant things surject to $K$-invariant, etc. No problem.
      $endgroup$
      – paul garrett
      Jan 7 at 18:14












      $begingroup$
      Thanks for your answer. I wrote an answer of my own explaining in my own words what you wrote. Would you mind seeing if there is anything incorrect about what I wrote? I want to make sure I understand your answer completely.
      $endgroup$
      – D_S
      Jan 8 at 3:49




      $begingroup$
      Thanks for your answer. I wrote an answer of my own explaining in my own words what you wrote. Would you mind seeing if there is anything incorrect about what I wrote? I want to make sure I understand your answer completely.
      $endgroup$
      – D_S
      Jan 8 at 3:49











      1












      $begingroup$

      To answer the question asked, yes, every discrete subgroup of $SL_2(mathbb R)$ has a fundamental domain, called a Dirichlet domain.






      share|cite|improve this answer









      $endgroup$


















        1












        $begingroup$

        To answer the question asked, yes, every discrete subgroup of $SL_2(mathbb R)$ has a fundamental domain, called a Dirichlet domain.






        share|cite|improve this answer









        $endgroup$
















          1












          1








          1





          $begingroup$

          To answer the question asked, yes, every discrete subgroup of $SL_2(mathbb R)$ has a fundamental domain, called a Dirichlet domain.






          share|cite|improve this answer









          $endgroup$



          To answer the question asked, yes, every discrete subgroup of $SL_2(mathbb R)$ has a fundamental domain, called a Dirichlet domain.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 7 at 3:07









          Lee MosherLee Mosher

          51.8k33889




          51.8k33889























              0












              $begingroup$

              Let $G = operatorname{SL}_2(mathbb R), K = operatorname{SO}_2(mathbb R)$, and $Gamma$ a discrete subgroup of $G$. We identify the measures $mu, dot mu, bar{mu}$ on $G, Gamma backslash G$ and $G/K$ respectively with positive linear functionals $T, dot T, bar{T}$ on $mathscr C_c(G), mathscr C_c(Gamma backslash G), mathscr C_c(G/K)$ respectively. For example,



              $$T(f) = intlimits_G f(g)dg tag{$f in mathscr C_c(G)$}$$



              $$bar{T}(f) = intlimits_{G/K} f(gK)dbar{mu}(gK) tag{$f in mathscr C_c(G/K)$}$$



              The upper half plane $mathbb H$ with the hyperbolic measure $frac{dxdy}{y^2}$ are identified with $G/K$ and $bar{mu}$, respectively.



              Since $K$ is compact, we have a natural identification of $mathscr C_c(G/K)$ with those elements of $mathscr C_c(G)$ which are right $K$-invariant, by precomposing with the quotient map $G rightarrow G/K$. We may similarly identify $mathscr C_c(Gammabackslash G/K)$ with the right $K$-invariant elements of $mathscr C_c(Gamma backslash G)$ via the quotient map $Gamma backslash G rightarrow Gamma backslash G/K$. The surjection



              $$delta: mathscr C_c(G) rightarrow mathscr C_c(Gamma backslash G)$$



              $$delta(f)(Gamma g) = sumlimits_{gamma in Gamma} f(gamma g)$$



              can be shown to restrict to a surjection



              $$mathscr C_c(G/K) rightarrow mathscr C_c(Gamma backslash G /K)$$



              The answer to my question follows if I can prove the following proposition.




              Proposition: There exists a unique linear positive functional $widetilde{T}$ on $mathscr C_c(Gamma backslash G/K)$, with corresponding measure $widetilde{mu}$, such that



              $$bar{T} = widetilde{T} circ delta$$




              In terms of measures, what this says is that if we define $widetilde{T}(f) =: intlimits_{Gamma backslash G/K} f(Gamma g K) dwidetilde{mu}(Gamma g K)$, then



              $$ intlimits_{G/K} f(gK) d bar{mu}(gK) = intlimits_{Gamma backslash G/K} space space spacebigg( sumlimits_{gamma in Gamma} space space f(gamma g K) bigg)space d widetilde{mu}(Gamma gK) $$



              or



              $$intlimits_{mathbb H} f(z)y^{-2} dxdy = intlimits_{Gamma backslash mathbb H} sumlimits_{gamma in Gamma} f(gamma.z) d widetilde{mu}(Gamma z)$$






              share|cite|improve this answer









              $endgroup$


















                0












                $begingroup$

                Let $G = operatorname{SL}_2(mathbb R), K = operatorname{SO}_2(mathbb R)$, and $Gamma$ a discrete subgroup of $G$. We identify the measures $mu, dot mu, bar{mu}$ on $G, Gamma backslash G$ and $G/K$ respectively with positive linear functionals $T, dot T, bar{T}$ on $mathscr C_c(G), mathscr C_c(Gamma backslash G), mathscr C_c(G/K)$ respectively. For example,



                $$T(f) = intlimits_G f(g)dg tag{$f in mathscr C_c(G)$}$$



                $$bar{T}(f) = intlimits_{G/K} f(gK)dbar{mu}(gK) tag{$f in mathscr C_c(G/K)$}$$



                The upper half plane $mathbb H$ with the hyperbolic measure $frac{dxdy}{y^2}$ are identified with $G/K$ and $bar{mu}$, respectively.



                Since $K$ is compact, we have a natural identification of $mathscr C_c(G/K)$ with those elements of $mathscr C_c(G)$ which are right $K$-invariant, by precomposing with the quotient map $G rightarrow G/K$. We may similarly identify $mathscr C_c(Gammabackslash G/K)$ with the right $K$-invariant elements of $mathscr C_c(Gamma backslash G)$ via the quotient map $Gamma backslash G rightarrow Gamma backslash G/K$. The surjection



                $$delta: mathscr C_c(G) rightarrow mathscr C_c(Gamma backslash G)$$



                $$delta(f)(Gamma g) = sumlimits_{gamma in Gamma} f(gamma g)$$



                can be shown to restrict to a surjection



                $$mathscr C_c(G/K) rightarrow mathscr C_c(Gamma backslash G /K)$$



                The answer to my question follows if I can prove the following proposition.




                Proposition: There exists a unique linear positive functional $widetilde{T}$ on $mathscr C_c(Gamma backslash G/K)$, with corresponding measure $widetilde{mu}$, such that



                $$bar{T} = widetilde{T} circ delta$$




                In terms of measures, what this says is that if we define $widetilde{T}(f) =: intlimits_{Gamma backslash G/K} f(Gamma g K) dwidetilde{mu}(Gamma g K)$, then



                $$ intlimits_{G/K} f(gK) d bar{mu}(gK) = intlimits_{Gamma backslash G/K} space space spacebigg( sumlimits_{gamma in Gamma} space space f(gamma g K) bigg)space d widetilde{mu}(Gamma gK) $$



                or



                $$intlimits_{mathbb H} f(z)y^{-2} dxdy = intlimits_{Gamma backslash mathbb H} sumlimits_{gamma in Gamma} f(gamma.z) d widetilde{mu}(Gamma z)$$






                share|cite|improve this answer









                $endgroup$
















                  0












                  0








                  0





                  $begingroup$

                  Let $G = operatorname{SL}_2(mathbb R), K = operatorname{SO}_2(mathbb R)$, and $Gamma$ a discrete subgroup of $G$. We identify the measures $mu, dot mu, bar{mu}$ on $G, Gamma backslash G$ and $G/K$ respectively with positive linear functionals $T, dot T, bar{T}$ on $mathscr C_c(G), mathscr C_c(Gamma backslash G), mathscr C_c(G/K)$ respectively. For example,



                  $$T(f) = intlimits_G f(g)dg tag{$f in mathscr C_c(G)$}$$



                  $$bar{T}(f) = intlimits_{G/K} f(gK)dbar{mu}(gK) tag{$f in mathscr C_c(G/K)$}$$



                  The upper half plane $mathbb H$ with the hyperbolic measure $frac{dxdy}{y^2}$ are identified with $G/K$ and $bar{mu}$, respectively.



                  Since $K$ is compact, we have a natural identification of $mathscr C_c(G/K)$ with those elements of $mathscr C_c(G)$ which are right $K$-invariant, by precomposing with the quotient map $G rightarrow G/K$. We may similarly identify $mathscr C_c(Gammabackslash G/K)$ with the right $K$-invariant elements of $mathscr C_c(Gamma backslash G)$ via the quotient map $Gamma backslash G rightarrow Gamma backslash G/K$. The surjection



                  $$delta: mathscr C_c(G) rightarrow mathscr C_c(Gamma backslash G)$$



                  $$delta(f)(Gamma g) = sumlimits_{gamma in Gamma} f(gamma g)$$



                  can be shown to restrict to a surjection



                  $$mathscr C_c(G/K) rightarrow mathscr C_c(Gamma backslash G /K)$$



                  The answer to my question follows if I can prove the following proposition.




                  Proposition: There exists a unique linear positive functional $widetilde{T}$ on $mathscr C_c(Gamma backslash G/K)$, with corresponding measure $widetilde{mu}$, such that



                  $$bar{T} = widetilde{T} circ delta$$




                  In terms of measures, what this says is that if we define $widetilde{T}(f) =: intlimits_{Gamma backslash G/K} f(Gamma g K) dwidetilde{mu}(Gamma g K)$, then



                  $$ intlimits_{G/K} f(gK) d bar{mu}(gK) = intlimits_{Gamma backslash G/K} space space spacebigg( sumlimits_{gamma in Gamma} space space f(gamma g K) bigg)space d widetilde{mu}(Gamma gK) $$



                  or



                  $$intlimits_{mathbb H} f(z)y^{-2} dxdy = intlimits_{Gamma backslash mathbb H} sumlimits_{gamma in Gamma} f(gamma.z) d widetilde{mu}(Gamma z)$$






                  share|cite|improve this answer









                  $endgroup$



                  Let $G = operatorname{SL}_2(mathbb R), K = operatorname{SO}_2(mathbb R)$, and $Gamma$ a discrete subgroup of $G$. We identify the measures $mu, dot mu, bar{mu}$ on $G, Gamma backslash G$ and $G/K$ respectively with positive linear functionals $T, dot T, bar{T}$ on $mathscr C_c(G), mathscr C_c(Gamma backslash G), mathscr C_c(G/K)$ respectively. For example,



                  $$T(f) = intlimits_G f(g)dg tag{$f in mathscr C_c(G)$}$$



                  $$bar{T}(f) = intlimits_{G/K} f(gK)dbar{mu}(gK) tag{$f in mathscr C_c(G/K)$}$$



                  The upper half plane $mathbb H$ with the hyperbolic measure $frac{dxdy}{y^2}$ are identified with $G/K$ and $bar{mu}$, respectively.



                  Since $K$ is compact, we have a natural identification of $mathscr C_c(G/K)$ with those elements of $mathscr C_c(G)$ which are right $K$-invariant, by precomposing with the quotient map $G rightarrow G/K$. We may similarly identify $mathscr C_c(Gammabackslash G/K)$ with the right $K$-invariant elements of $mathscr C_c(Gamma backslash G)$ via the quotient map $Gamma backslash G rightarrow Gamma backslash G/K$. The surjection



                  $$delta: mathscr C_c(G) rightarrow mathscr C_c(Gamma backslash G)$$



                  $$delta(f)(Gamma g) = sumlimits_{gamma in Gamma} f(gamma g)$$



                  can be shown to restrict to a surjection



                  $$mathscr C_c(G/K) rightarrow mathscr C_c(Gamma backslash G /K)$$



                  The answer to my question follows if I can prove the following proposition.




                  Proposition: There exists a unique linear positive functional $widetilde{T}$ on $mathscr C_c(Gamma backslash G/K)$, with corresponding measure $widetilde{mu}$, such that



                  $$bar{T} = widetilde{T} circ delta$$




                  In terms of measures, what this says is that if we define $widetilde{T}(f) =: intlimits_{Gamma backslash G/K} f(Gamma g K) dwidetilde{mu}(Gamma g K)$, then



                  $$ intlimits_{G/K} f(gK) d bar{mu}(gK) = intlimits_{Gamma backslash G/K} space space spacebigg( sumlimits_{gamma in Gamma} space space f(gamma g K) bigg)space d widetilde{mu}(Gamma gK) $$



                  or



                  $$intlimits_{mathbb H} f(z)y^{-2} dxdy = intlimits_{Gamma backslash mathbb H} sumlimits_{gamma in Gamma} f(gamma.z) d widetilde{mu}(Gamma z)$$







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Jan 8 at 3:47









                  D_SD_S

                  14.2k61754




                  14.2k61754






























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