Wednesday, 17 October 2018
Ilir Snopche: Test elements and retracts in free groups
An element \(x\) of a group \(G\) is called a test element if for any endomorphism \(\phi\) of \(G\), \(\phi(x) = x\) implies that \(\phi\) is an automorphism. The first non-trivial example of a test element was given by Nielsen in 1918, when he proved that every endomorphism of a free group of rank 2 that fixes the commutator \([x, y]\) of a pair of generators must be an automorphism. A subgroup \(R\) of a group \(G\) is said to be a retract of \(G\) if there is a homomorphism \(r : G \to R\) that restricts to the identity on \(R\).In this talk I will discuss the distribution of test elements in free groups and the role of retracts in the study of endomorphisms of free groups. This is a joint work with Slobodan Tanushevski.