An associate of Meyer Lansky, Greenbaum joined his organization on New York's Lower East Side sometime during the mid to late 1910s. During Prohibition, Greenbaum began working with the Chicago Outfit managing the southwest division of the Trans-America wire service in 1928. Sent to Las Vegas shortly after WWII, Greenbaum gained control over syndicate gambling operations, with Morris Rosen and Moe Sedway, following the death of Bugsy Siegel in 1947.
Taking over as manager of the Flamingo Hotel, Greenbaum brought the struggling casino out of debt within several months, eventually controlling several other syndicate casinos and bookmaking operations in Arizona within several years. As a leading syndicate leader in Las Vegas, Greenbaum would later order the deaths of Tony Brancato and Tony Tombino after robbing a syndicate hotel. Shortly after becoming manager of the Riviera Hotel, Greenbaum's excessive gambling, womanizing, and drug habits eventually caused him to begin skimming from casino operations. However, his embezzlement was soon discovered by the Chicago syndicate, and on December 3, 1958, both he and his wife were found in their Phoenix home with their throats slashed.