A battle within the Republican Party over same-sex marriage is unfolding on two fronts, in public, and behind the scenes. In the latter case, one of the most influential players is a billionaire hedge fund manager largely unknown to those who don’t work in finance or mix with political mega-donors. That man is Paul E. Singer, who over the years has used his wealth to spur Republicans to support gay marriage laws. Now, Singer is expanding his reach with the creation of an advocacy group which aims to spend millions influencing the legislative debate over same-sex marriage across the country.