Irwin Kra, Automorphic Forms and Kleinian Groups (Benjamin, 1972),xiv+464 pp.

Discontinuous Groups and Automorphic Forms

Automorphic Forms and Metaplectic Groups

In 1952, Gelfand and Fomin noticed that classical modular forms were related to representations of SL2(R). As a result of this realization, Gelfand later defined GLr automorphic forms via representation theory. A metaplectic form is just an automorphic form defined on a cover of GLr, called a metaplectic group. In this talk, we will carefully construct the metaplectic covers of GL2(F) where F i...

Kleinian Groups, Laplacian on Forms and Currents at Infinity

In this note we consider the spectrum of the Laplacian acting on the space of (co-closed) differential forms on the quotient of /i-dimensional hyperbolic space by a co-compact Kleinian group. Using a result of P.-Y. Gaillard we relate these to currents on the sphere at infinity of hyperbolic space with distinctive transformation properties under the action of the group. We analyse these current...

Automorphic functions for a Kleinian group

In the paper ‘Automorphic functions for a Whitehead-complement group’ [5], Matsumoto, Nishi and Yoshida constructed automorphic functions on real 3–dimensional hyperbolic space for a Kleinian group called the Whitehead-link-complement group. For a Kleinian group (of the first kind), no automorphic function/form has been studied before. In this note, their motivation is presented with a historic...

Cohomology of Arithmetic Groups and Periods of Automorphic Forms

(1) Firstly, I gave a brief introduction to the cohomology of arithmetic groups, in particular the fact that the “tempered” (informally: “near the middle dimension”’) part of the cohomology looks like the cohomology of a torus. This corresponds to §2, §3, §4 of these notes. For the reader totally unfamiliar with the cohomology of arithmetic groups, §3 might be the best thing to focus on. (2) Ne...