How do you compute the Fundamental group of the connected sum of infinitely many tori?

I know the Fund. Group of the connected sum of n tori is given by

$\displaystyle \pi _1 (T\# T\# \cdots \# T)= < \beta _1 , \gamma _1 , \cdots , \beta _n , \gamma _n / \beta _1 \gamma _1 \beta^{-1} _1 \gamma^{-1} _1 \cdots \beta _n \gamma _n \beta^{-1} _n \gamma^{-1} _n =1> $.

But what about the Fundamental group of the surface given by the connected sum of infinitely many tori, and how do you prove this?