9 problems

Let $F$ be a free group. For each positive integer $n$, let $U_n$ be the set of all non-minimal elements in $F$ of length $n$, and let $NU_n\subset U_n$ be the subset consisting of…

Let $F$ be a free group, and let a non-minimal element of $F$ be represented by its corresponding Whitehead Graph. A Nielsen automorphism is an automorphism of the type used in Whi…

Let $F_r$ be a free group, let $NMin\subset F_r$ be the set of non-minimal elements, and let $WC(w)$ denote Whitehead complexity. Polynomial Whitehead-descent time-complexity conje…

Let $F_r$ be a free group, and let $T=T_r$ be the restricted set of Whitehead automorphisms described in the source. For each length $l$, let $NMin_l\subset F_r$ be the set of non-…

Exponential word-length growth conjecture. There exists a constant $c_r>1$ such that

Let $F=F_n$ be a free group of rank $n$, and let $NMin_l\subset F$ be the set of all non-minimal elements of length $l$. Let $LR(w)$ denote the length-reducing complexity associate…

Let $F_r$ be a free group of rank $r\geq 2$, and interpret “almost all” using asymptotic density on the set of elements of $F_r$ ordered by word length. An element is Whitehead min…

Let $F_r$ be a free group of rank $r$, let $F_r^k$ denote the set of $k$-tuples of elements of $F_r$, and let $U_{\min}$ be a minimal representative of the orbit of $U$ under the r…

Let $F_r$ be a free group, and let $w\in F_r$ be a word whose length cannot be reduced by any elementary Whitehead automorphism; in particular, $w$ is cyclically reduced. Consider…