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Let $F_n$ be the free group with $n$ generators $\gamma_1,\dots,\gamma_n$. Consider the homomorphisms $h_i\colon F_n\to F_{n-1}$ defined by adding the relation $\gamma_i=1$ in $F_n$.

What is the least word norm of a nontrivial element $\gamma\in F_n$ such that $h_i(\gamma)=e$ for any $i$?

Let $F_n$ be the free group with $n$ generators $\gamma_1,\dots,\gamma_n$. Consider the homomorphisms $h_i\colon F_n\to F_{n-1}$ defined by adding the relation $\gamma_i=1$ in $F_n$.

What is the least word norm of a nontrivial element $\gamma\in F_n$ such that $h_i(\gamma)=1$ for each $i$?

Let $F_n$ be the free group with $n$ generators $\gamma_1,\dots,\gamma_n$. Consider the homomorphisms $h_i\colon F_n\to F_{n-1}$ defined by adding the relation $\gamma_i=1$ in $F_n$.

What is the least word norm of a nontrivial element $\gamma\in F_n$ such that $h_i(\gamma)=e$ for any $i$?

Postscript. The question seems to be open, see Picture-Hanging Puzzles by Erik D. Demaine, Martin L. Demaine, Yair N. Minsky, Joseph S. B. Mitchell, Ronald L. Rivest, and Mihai Patrascu.

Let $F_n$ be the free group with $n$ generators $\gamma_1,\dots,\gamma_n$. Consider the homomorphisms $h_i\colon F_n\to F_{n-1}$ defined by adding the relation $\gamma_i=1$ in $F_n$.

What is the least word norm of a nontrivial element $\gamma\in F_n$ such that $h_i(\gamma)=e$ for any $i$?

Let $F_n$ be the free group with $n$ generators $\gamma_1,\dots,\gamma_n$. Consider the homomorphisms $h_i\colon F_n\to F_{n-1}$ defined by adding the relation $\gamma_i=1$ in $F_n$.

What is the least word norm of a nontrivial element $\gamma\in F_n$ such that $h_i(\gamma)=e$ for any $i$?

Let $F_n$ be the free group with $n$ generators $\gamma_1,\dots,\gamma_n$. Consider the homomorphisms $h_i\colon F_n\to F_{n-1}$ defined by adding the relation $\gamma_i=1$ in $F_n$.

What is the least word norm of a nontrivial element $\gamma\in F_n$ such that $h_i(\gamma)=e$ for any $i$?

Postscript. The question seems to be open, see Picture-Hanging Puzzles by Erik D. Demaine, Martin L. Demaine, Yair N. Minsky, Joseph S. B. Mitchell, Ronald L. Rivest, and Mihai Patrascu.