18.42 (2014)

For an orientation-preserving homeomorphism $f : S^1 \to S^1$ of a unit circle $S^1$, let $\tilde{f}$ be its lifting to a homeomorphism of $\mathbb{R}$; then the rotation number of $f$ is defined to be the limit $\lim_{n \to \infty}(\tilde{f}^n(x) - x)/n$ (which is independent of the point $x \in S^1$). Let $F$ be a free group of rank 2 with generators $a, b$. For $w \in F$ and $r, s \in \mathbb{R}$, let $R(w, r, s)$ denote the maximum value of the (real-valued) rotation number of $w$, under all representations from $F$ to the universal central extension of $\text{Homeo}^+(S^1)$ for which the rotation number of $a$ is $r$, and the rotation number of $b$ is $s$.
$\qquad$ a) If $r, s$ are rational, must $R(w, r, s)$ be rational?
$\qquad$ b) Let $R(w, r^-, s^-)$ denote the supremum of the rotation number of $w$ (as above) under all representations for which $a$ and $b$ are conjugate to rotations through $r$ and $s$, respectively (in the universal central extension of $\text{Homeo}^+(S^1)$). Is $R(w, r^-, s^-)$ always rational if $r$ and $s$ are rational?
$\qquad$ c) Weak Slippery Conjecture: Let $w$ be a word containing only positive powers of $a$ and $b$. A pair of values $(r, s)$ is slippery if there is a strict inequality $R(w, r', s') < R(w, r^-, s^-)$ for all $r' < r$, $s' < s$. Is it true that then $R(w, r^-, s^-) = h_a(w)r + h_b(w)s$, where $h_a(w)$ counts the number of $a$’s in $w$, and $h_b(w)$ counts the number of $b$’s in $w$?
$\qquad$ d) Slippery Conjecture: More precisely, is it always (without assuming $(r, s)$ being slippery) true that if $w$ has the form $w = a^{\alpha_1} b^{\beta_1} \dots a^{\alpha_m} b^{\beta_m}$ (all positive) and $R(w, r, s) = p/q$ where $p/q$ is reduced, then $|p/q - h_a(w)r - h_b(w)s| \leqslant m/q$?