Some simple questions regarding Modular Spaces, Modular Forms, Hyperbolic Geometry, Projective Special Linear and Modular Groups

dynamical systemsgeodesichyperbolic-geometrymodular-formsquotient-spaces

Asymptotic Winding of the Geodesic Flow on Modular Surfaces and Continuous Fractions. Y. Guivarc'h and Y. Le Jan.
enter image description here

I have been struggling with the first section (1. Framework and notations) for a long time.

I have some questions, that are:

  1. How is the hyperbolic distance defined to get that the hyperbolic distance between $gK$ and $hK$ is $2 \log(||h^{-1}g||)$?

  2. what do they mean by saying $||\cdot||$ is the euclidean operator norm? How is it defined?

  3. the last line we have $i(c+d)^{-2}$, is not it supposed to be $i(ci+d)^{-2}$?

  4. what is the idea of the last line?
    enter image description here

  5. Is $C_K(G)$ the space of all complex-valued continuous maps with compact support defined on $G$?

  6. after how Haar measure is defined, is not it supposed to be for any $g$ such that $g(0)= \alpha$ not $g(0)= \infty$?

  7. $\Sigma$ is finite, Does it mean it is finite with respect to a measure or finite in the usual sense that it is cardinality is finite?

  8. Are not cusps subsets of $\mathbb{Q} \cup\infty$?

  9. How does $\Sigma$ act transitively on $C$?

  10. How is the stabiliser of the cusp defined?

  11. What is the order of the stabiliser?

  12. The space of entire modular forms of dimension $-2$, How is it possible (negative dimension)?
    enter image description here

enter image description here
In the end, I would like to enquire regarding the references that might help me understand the first section properly, regardless of being the references of this article or not. (In the worst case, I would be very grateful if one could tell me the references that I need from the image attached below).
enter image description here
enter image description here

Best Answer

(First, this is a somewhat awkward way to present your question... giving us no idea of your personal context, for example, ...)

All that your copied text discusses is absolutely standard "elliptic modular forms, Riemann surfaces" stuff. R. Gunning's classic orange Princeton series book on "modular forms" is terrific.

Or nearly any other of the many introductions in either texts or on-line in the last 60+ years. E.g., my course notes at my web page.

It's not the sort of thing that one absorbs in a few minutes or a few days... And, no, many people have never heard of any of it, etc. But, in fact, this sort of thing is a huge part of contemporary mathematics. Played a role in A. Wiles' proof of Fermat's Last Theorem, and such.

No wonder that a typical undergrad math preparation does not prepare a person for this sort of thing.

Related Question