ICTS

In this talk, we explore the following question about words in free groups: if two words induce identical images as word maps on every finite group, must they be endomorphic to each other? This leads to the concept of weak profinite rigidity, a property analogous to profinite rigidity of words. We illustrate this distinction with the example of the words x1 and x²₁x₂x₁⁻¹x₂⁻¹ which have equal images but distinct measures on some finite group. Finally, we present recent progress establishing weak profinite rigidity for some words.