## Jacobi Group on $H \times C$

Hi,

I am reading the book Eichler, Zagier about the theory of Jacobi forms and have a question regarding page 13

"In the situation of modular forms, we write $H$ as $G / K$, where $G = SL_2(R)$ contains $\Gamma$ as a discrete subgroup with $Vol( \Gamma \backslash G)$ finite and $K = SO(2)$ is a maximal compacting subgroup of $G$. Here we would like to do the same. However, the group $SL_2(R) \ltimes H_R$ contains $\Gamma^J = \Gamma\times Z^2$ with infinite covolume (because of the extra $R$ in $H_R$) and its quotient by the maximal compacting subgroup $SO(2)$ is $H \times C \times R$ rather than $H\times C$."

After spending several hours (actually already several days on the Jacobi groups), I still have no clue why

"Here we would like to do the same. However, the group $SL_2(R)\ltimes H_R$ contains $\Gamma^J = \Gamma\times Z^2$ with infinite covolume (because of the extra $R$ in $H_R$) and its quotient by the maximal compacting subgroup $SO(2)$ is $H\times C\times R$ rather than $H\times C$."

I assume that the Heisenberg group adds the extra $R$. However, I don't know the rules why this is the case.

 I couldn't stand the use of X instead of $\times$, so I changed them (and accidentally hit enter while summarising this edit). – David Roberts Jan 19 2011 at 6:06