One would want to look at the etymological trees of the words 'slave' and 'share' to see if there are any intersections or common ancestors.

Unfortunately I don't know of any resources that have that capability. Do you?

Well, they don't. "Slave" comes from the Old Church Slavonic word for Slavs, "Sloveninu," which is probably from the word "slovo," meaning speech, while "share" ultimately comes from the Proto Indo-European root "skar-," meaning to cut.

Dictionary.com says that "share" is from German, and "slave" is from Latin. No common origin.