To see many ideas that are not here, please read the paper published by D. Borthwick. The intention to this post is not to add anything new neither prove anything else, but only to present new things about resonances on hyperbolic surfaces. Maybe I’ve not beginning with something very basic, however, I’m planing to do a kind a state of art about these topics in the future.
Roughly speaking, a Schottky group is a particular kind of a Fuchsian group with the following construction.
Let be a set of disks such that exists a transformation that sends to , in this way, maps the exterior of to the interior of , where is defined cyclically, this means
A Schottky group is generated by the corresponding ’s. The domain for the action of is since each transformation is hyperbolic in the sense that it’s repelling a point inside and attracting a point inside .
Futhermore, if we take the closures of the disks are mutually disjoints means that has infinite area with no cusps, so is convex cocompact.
Selberg Zeta Function
It’s analogous to the Riemann zeta function, described as
where run all over the prime congruent class and is the norm of congruent class .
There is a linear operator for a set . We define the transfer operation as an operator such that
where is acting on the space of functions and is an auxiliary valuation function.
Zeta function for Schottky groups
In the paper Distribution of resonances for hyperbolic surfaces by D. Borthwick, there is a full section dedicated to this. Let be a map such that where is the union of the disks. The tranfer operator, acting over the Hilbert space and define
And then thanks to that, we can use it to write the Selberg zeta function as a determinant for all