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.

Schottky group

Roughly speaking, a Schottky group is a particular kind of a Fuchsian group with the following construction.

Let $\{D_{i}\}$ be a set of disks such that exists a transformation $S_{i} \in \psl{2}{\RR}$ that sends $\partial D_{i}$ to $\partial D_{i+r}$, in this way, maps the exterior of $D_{i}$ to the interior of $D_{i+r}$, where $i$ is defined cyclically, this means

A Schottky group $\Gamma$ is generated by the corresponding $S_{i}$’s. The domain for the action of $\Gamma$ is $\HH - \cup D_{i}$ since each transformation $S_{i}$ is hyperbolic in the sense that it’s repelling a point inside $D_{i}$ and attracting a point inside $D_{i+r}$.

Futhermore, if we take the closures of the disks are mutually disjoints means that $X \cong \Gamma \backslash \HH$ has infinite area with no cusps, so $\Gamma$ is convex cocompact.

Selberg Zeta Function

It’s analogous to the Riemann zeta function, described as

where $p$ run all over the prime congruent class and $N(p)$ is the norm of congruent class $p$.

Transfer operation

There is a linear operator $f: X \longrightarrow X$ for a set $X$. We define the transfer operation as an operator $\mathfrak{L}$ such that

where $\mathfrak{L}$ is acting on the space of functions $\{\phi: X \longrightarrow \CC \}$ and $g: X \longrightarrow \CC$ 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 $B$ be a map such that $B: U \longrightarrow \CC \cup \{\infty\}$ where $U = \bigcup_{j=1}^{2r} D_{j}$ is the union of the disks. The tranfer operator, acting over the Hilbert space $\mathcal{H}(U)$ and define

And then thanks to that, we can use it to write the Selberg zeta function as a determinant for all $s \in \CC$