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YCor
YCor
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Let $F_n$ be a free group of rank n and consider all possible non-degenerate length functions on $F_n$.

Could I be directed to a reference that give two non-trivial non-power-conjugate elements $g,h \in F_n$ (i.e. no power of $g$ is conjugate to a power of $h$) such that there is some $K>1$ so that for any length function we have: $$ ||g||_T \geq K \cdot ||h||_T, $$$$ \|g\|_T \geq K \cdot \|h\|_T, $$ where $||\cdot||_T$$\|\cdot\|_T$ denote translation length for some $T \in CV_n$$T \in \operatorname{CV}_n$? Obviously, we have this if $h^K=g$, but I specifically want a pair of elements that are not powers of one another.

Let $F_n$ be a free group of rank n and consider all possible non-degenerate length functions on $F_n$.

Could I be directed to a reference that give two non-trivial non-power-conjugate elements $g,h \in F_n$ (i.e. no power of $g$ is conjugate to a power of $h$) such that there is some $K>1$ so that for any length function we have: $$ ||g||_T \geq K \cdot ||h||_T, $$ where $||\cdot||_T$ denote translation length for some $T \in CV_n$? Obviously, we have this if $h^K=g$, but I specifically want a pair of elements that are not powers of one another.

Let $F_n$ be a free group of rank n and consider all possible non-degenerate length functions on $F_n$.

Could I be directed to a reference that give two non-trivial non-power-conjugate elements $g,h \in F_n$ (i.e. no power of $g$ is conjugate to a power of $h$) such that there is some $K>1$ so that for any length function we have: $$ \|g\|_T \geq K \cdot \|h\|_T, $$ where $\|\cdot\|_T$ denote translation length for some $T \in \operatorname{CV}_n$? Obviously, we have this if $h^K=g$, but I specifically want a pair of elements that are not powers of one another.

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NWMT
NWMT
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Pairs of elements in $F_n$ with distinct translation lengths

Let $F_n$ be a free group of rank n and consider all possible non-degenerate length functions on $F_n$.

Could I be directed to a reference that give two non-trivial non-power-conjugate elements $g,h \in F_n$ (i.e. no power of $g$ is conjugate to a power of $h$) such that there is some $K>1$ so that for any length function we have: $$ ||g||_T \geq K \cdot ||h||_T, $$ where $||\cdot||_T$ denote translation length for some $T \in CV_n$? Obviously, we have this if $h^K=g$, but I specifically want a pair of elements that are not powers of one another.