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Fundamental region for the modular group
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Timothy Murphy
science forum Guru Wannabe


Joined: 29 Apr 2005
Posts: 275

PostPosted: Mon Jul 17, 2006 8:32 pm    Post subject: Fundamental region for the modular group Reply with quote

This is a rather vague question ...
The modular group PSL(2,Z) acts on the upper half-plane H.
Let F be the usual fundamental region,
bounded by the lines Rl(z) = +/- 1/2,
and the portion of the circle |z| = 1 between omega and -omega^2
(with some identitification of boundary points).

As I understand it, there is a fairly natural way
of turning F + {infty} into a Riemann surface, F* say.

[I think this surface has genus 0,
but my colleague Dmitri, who is nearly always right -
and always convinced he is right - says it is of genus 2.
But that is irrelevant.]

One can also complete H + {infty} + Q (the rationals)
to a Riemann surface, H* say,
which is (probably) the universal cover of the first surface.

Am I right so far?
If so, my question is: Is this construction unique?
And could F* be derived directly from the group,
without this clever construction?




--
Timothy Murphy
e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
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Rupert
science forum Guru


Joined: 18 May 2005
Posts: 372

PostPosted: Mon Jul 17, 2006 11:13 pm    Post subject: Re: Fundamental region for the modular group Reply with quote

Timothy Murphy wrote:
Quote:
This is a rather vague question ...
The modular group PSL(2,Z) acts on the upper half-plane H.
Let F be the usual fundamental region,
bounded by the lines Rl(z) = +/- 1/2,
and the portion of the circle |z| = 1 between omega and -omega^2
(with some identitification of boundary points).

As I understand it, there is a fairly natural way
of turning F + {infty} into a Riemann surface, F* say.

[I think this surface has genus 0,
but my colleague Dmitri, who is nearly always right -
and always convinced he is right - says it is of genus 2.
But that is irrelevant.]

One can also complete H + {infty} + Q (the rationals)
to a Riemann surface, H* say,
which is (probably) the universal cover of the first surface.


Can you elaborate on this?

Quote:
Am I right so far?
If so, my question is: Is this construction unique?
And could F* be derived directly from the group,
without this clever construction?




--
Timothy Murphy
e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
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Timothy Murphy
science forum Guru Wannabe


Joined: 29 Apr 2005
Posts: 275

PostPosted: Tue Jul 18, 2006 11:01 am    Post subject: Re: Fundamental region for the modular group Reply with quote

Rupert wrote:

Quote:
This is a rather vague question ...
The modular group PSL(2,Z) acts on the upper half-plane H.
Let F be the usual fundamental region,
bounded by the lines Rl(z) = +/- 1/2,
and the portion of the circle |z| = 1 between omega and -omega^2
(with some identitification of boundary points).

As I understand it, there is a fairly natural way
of turning F + {infty} into a Riemann surface, F* say.

[I think this surface has genus 0,
but my colleague Dmitri, who is nearly always right -
and always convinced he is right - says it is of genus 2.
But that is irrelevant.]

One can also complete H + {infty} + Q (the rationals)
to a Riemann surface, H* say,
which is (probably) the universal cover of the first surface.

Can you elaborate on this?

Well, I'm really asking if this is true.
As I understand it, one has to add a cusp at each rational point a/b
to match the cusp at infty,
since PSL(2,Z) can take infty to any such point.

Assuming that one can construct a Riemann surface in this way -
and I am asking among other questions if one can -
then it would seem to me that any loop can be slightly perturbed
to miss any cusp, and then it would seem the space is simply-connected,
and since it covers what I called F* it must be the universal covering.

I hasten to add that I am not an expert or even knowledgeable
in this area, and was asking if I was correct,
not suggesting that I was!

--
Timothy Murphy
e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
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Lee Rudolph
science forum Guru


Joined: 28 Apr 2005
Posts: 566

PostPosted: Tue Jul 18, 2006 11:43 am    Post subject: Re: Fundamental region for the modular group Reply with quote

Timothy Murphy <tim@birdsnest.maths.tcd.ie> writes:

Quote:
This is a rather vague question ...
The modular group PSL(2,Z) acts on the upper half-plane H.
Let F be the usual fundamental region,
bounded by the lines Rl(z) = +/- 1/2,
and the portion of the circle |z| = 1 between omega and -omega^2
(with some identitification of boundary points).

As I understand it, there is a fairly natural way
of turning F + {infty} into a Riemann surface, F* say.

[I think this surface has genus 0,

You think right. More specifically, F* is homeomorphic to
the 2-sphere, but it is not (in the raw) smooth at its
so-called "cusps", those being the point covered by the
two points on the circle, and the point at infinity.
However, the conformal structure on the fundamental domain
immediately gives a conformal structure to all of F*
including the cusps (essentially because the maps z -> z^2
and z -> z^3 are conformal where z is non-zero) which then
lets you smooth the whole thing, making it a Riemann surface.

Quote:
but my colleague Dmitri, who is nearly always right -
and always convinced he is right - says it is of genus 2.
But that is irrelevant.]

Hmmph.

Quote:
One can also complete H + {infty} + Q (the rationals)
to a Riemann surface, H* say,
which is (probably) the universal cover of the first surface.

Am I right so far?

You cannot be, on your own terms: as you (and I) have defined
it, F* is compact, and it is of genus 0, so it's a 2-sphere,
thus simply-connected; so it's its own universal cover.

On the other hand, in the wonderful world of "orbifolds"
(Satake V-manifolds), I think that F* -- considered as
a sphere with two distinguished points, one labelled "3"
and one labelled "2" -- has an *orbifold* "universal cover"
that is indeed the (underlying orbifold of) H*.

Quote:
If so, my question is: Is this construction unique?
And could F* be derived directly from the group,
without this clever construction?

You would prefer a stupid construction, then?

Anyway, yes, F* can be "derived directly from the group"
in other ways, which cannot however be guaranteed not to
be clever. The point is (I think; presumably Derek Holt
will kindly correct any extravagantly wrong statements in
what I'm about to say, unless indeed it is so confused that
there'd be no point at all in trying to salvage anything)
that G = PSL(2,Z) has non-trivial torsion, and therefore
the Eilenberg-MacLane space K(G,1) cannot be finite-dimensional;
however, the torsion subgroup T has finite index (12? or 6?
that P is confusing me) in G, and G/T *does* have a finite-dimensional
Eilenberg-MacLane space. In fact, G/T has cohomological dimension 2,
and among the various explicit constructions of K(pi,1)'s at
least one (which begins, I think, with a sufficiently nice
presentation of the group pi), when applied to G/T, produces
a simplicial complex M that not only is of *geometric* dimension 2,
but is actually a 2-manifold ... of (maybe) genus 2 (which, if so,
might explain your colleague's confusion described earlier).
Now T acts, *with fixed points*, on M; and the complex M/T
is homeomorphis to F*.

....I have a very bad feeling about that last paragraph, but instead
of trying to make it right I will leave my shame on public view in
the hope that it will attract a speedy and accurate correction.
Meanwhile I will have my first caffeine of the day while I wait
for developments.

Lee Rudolph
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Timothy Murphy
science forum Guru Wannabe


Joined: 29 Apr 2005
Posts: 275

PostPosted: Tue Jul 18, 2006 6:33 pm    Post subject: Re: Fundamental region for the modular group Reply with quote

Lee Rudolph wrote:

Quote:
If so, my question is: Is this construction unique?
And could F* be derived directly from the group,
without this clever construction?

You would prefer a stupid construction, then?

I'm not sure the alternative construction is stupid enough for me.
But thank you very much for your response,
which was most enlightening;
I shall certainly pursue the points you raise.

--
Timothy Murphy
e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
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Rupert
science forum Guru


Joined: 18 May 2005
Posts: 372

PostPosted: Wed Jul 19, 2006 12:14 am    Post subject: Re: Fundamental region for the modular group Reply with quote

Timothy Murphy wrote:
Quote:
Rupert wrote:

This is a rather vague question ...
The modular group PSL(2,Z) acts on the upper half-plane H.
Let F be the usual fundamental region,
bounded by the lines Rl(z) = +/- 1/2,
and the portion of the circle |z| = 1 between omega and -omega^2
(with some identitification of boundary points).

As I understand it, there is a fairly natural way
of turning F + {infty} into a Riemann surface, F* say.

[I think this surface has genus 0,
but my colleague Dmitri, who is nearly always right -
and always convinced he is right - says it is of genus 2.
But that is irrelevant.]

One can also complete H + {infty} + Q (the rationals)
to a Riemann surface, H* say,
which is (probably) the universal cover of the first surface.

Can you elaborate on this?

Well, I'm really asking if this is true.

I doubt it. How are you going to do the completion?
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