CARMA COLLOQUIUM Speaker: Assoc Prof Murray Elder, CARMA, The University of Newcastle Title: Random subgroups of Thompson's group F Location: Room V129, Mathematics Building (Callaghan Campus) The University of Newcastle Time and Date: 4:00 pm, Thu, 18th Nov 2010 Abstract: What is a {\em random subgroup} of a group, and who cares? In a (non-abelian) group based cryptosystem, two parties (Alice and Bob) each choose a subgroup of some platform group "at random" -- each picks $k$ elements "at random" and takes the subgroup generated by their chosen elements. But for some platform groups (like the braid groups, which were chosen first, being so complicated and difficult) a "random subgroup" is not so random after all. It turned out, pick $k$ elements of a braid group, and they will generate (almost always) a {\em free group} with your $k$ elements as the free basis. And if Alice and Bob are just playing with free groups, it makes their secrets easy to attack. Richard Thompson's group $F$ is an infinite, torsion free group, with many weird and cool properties, but the one I liked for this project is that it has {\em no} free subgroups (of rank $>1$) at all, so a random subgroup of $F$ could not be free -- so what would it be? This is joint work with Sean Cleary (CUNY), Andrew Rechnitzer (UBC) and Jeniffer Taback (Bowdoin). [Permanent link]