No small subgroups vs no small normal subgroups.

A topological group is said to have no small subgroup (resp. no small normal subgroup) if it admits an open neighbourhood of the identity containing no non-trivial subgroup (resp. normal subgroup) of . These properties are usually denoted by (and respectively ). The property plays an important historical role in the solution to the fifth problem of Hilbert due to Gleason, Montgomery-Zippin and Yamabe for the characterization of Lie groups.

In this talk we present an example of a free group (with countably infinitely many generators) which is (through topological simplicity), but which admits no non-trivial continuous homomorphism to groups; showing a very strong contrast between both properties.