Explicit solutions of certain orientable quadratic equations in free groups

Abstract

For $g\geq1$ denote by $F_{2g}=\langle x_1, y_1,\dots,x_g,y_g\rangle$ the free group on $2g$ generators and by $B_g=[x_1,y_1]\dots[x_g,y_g]$. For $l,c\geq 1$ and elements $w_1,\dots,w_l\in F_{2g}$ we study orientable quadratic equations of the form $[u_1,v_1]\dots[u_h,v_h]=(B_g^{w_1})^c(B_g^{w_2})^c\dots(B_g^{w_l})^c$ with unknowns $u_1,v_1,\dots,u_h,v_h$ and provide explicit solutions for them for the minimal possible number $h$. In the particular case when $g=1$, $w_i=y_1^{i-1}$ for $i=1,\dots,l$ and $h$ the minimal number which satisfies $h \geq l(c-1)/2+1$ we provide two types of solutions depending on the image of the subgroup $H=\langle u_1,v_1,\dots,u_h,v_h\rangle$ generated by the solution under the natural homomorphism $p:F_2\to F_2/[F_2,F_2]$: the first solution, which is called a primitive solution, satisfies $p(H)=F_2/[F_2,F_2]$, the second solution satisfies $p(H) = \big\langle p(x_1),p(y_1^l)\big\rangle$. We also provide an explicit solution of the equation $[u_1,v_1]\dots[u_k, v_k] = \big(B_1\big)^{k+l} \big({B_1}^{y}\big)^{k-l}$ for $k>l\geq0$ in $F_2$, and prove that if $l\neq0$, then every solution of this equation is primitive. As a geometrical consequence, for every solution we obtain a map $f:S_h\to T$ from the orientable surface $S_h$ of genus $h$ to the torus $T=S_1$ which has the minimal number of roots among all maps from the homotopy class of $f$. Depending on the number $|p(F_2):p(H)|$ such maps have fundamentally different geometric properties: in some cases they satisfy the Wecken property and in other cases not.