Uploaded June 2026 | Updated September 2026, 3 days ago
The 25th Midrasha Mathematicae on Groups, Expanders and Codes -- Celebrating Alex Lubotzky's 70th birthday.
Day 4, Session 1
Speaker: Yotam Shomroni (Weizmann)
Title: Independent Words are Freely Separated
Abstract: Every word w in a free group F defines, for each finite group G, a G-valued random variable, by substitution of independent, uniformly random G-elements in the letters. If w_1 and w_2 share no letters, they define independent random variables. We prove the converse: if w_1 and w_2 are independent for every finite G, then they share no letters when written in some basis of F. In fact, it suffices that w_1 and w_2 are independent in the symmetric groups Sn; we show that the correlation between the events of w_i fixing some point is non-negative for all large enough n, and positive unless the words share no letters (in any basis).More generally, for finitely generated subgroups H_1, ..., H_k of F, the random subgroups they define in a finite group G are independent for every G, if and only if some basis of F supports them on disjoint letters.These results build upon the seminal work of Puder--Parzanchevski, of similar spirit, which showed that measure preserving words are primitive.
The 25th Midrasha Mathematicae on Groups, Expanders and Codes -- Celebrating Alex Lubotzky's 70th birthday.
Day 4, Session 1
Speaker: Yotam Shomroni (Weizmann)
Title: Independent Words are Freely Separated
Abstract: Every word w in a free group F defines, for each finite group G, a G-valued random variable, by substitution of independent, uniformly random G-elements in the letters. If w_1 and w_2 share no letters, they define independent random variables. We prove the converse: if w_1 and w_2 are independent for every finite G, then they share no letters when written in some basis of F. In fact, it suffices that w_1 and w_2 are independent in the symmetric groups Sn; we show that the correlation between the events of w_i fixing some point is non-negative for all large enough n, and positive unless the words share no letters (in any basis).More generally, for finitely generated subgroups H_1, ..., H_k of F, the random subgroups they define in a finite group G are independent for every G, if and only if some basis of F supports them on disjoint letters.These results build upon the seminal work of Puder--Parzanchevski, of similar spirit, which showed that measure preserving words are primitive.
