Explicit solutions of certain orientable quadratic equations in free groups

TL;DR

This paper provides explicit solutions to certain orientable quadratic equations in free groups, analyzing their algebraic structure and geometric implications, including minimal maps from surfaces to tori and properties like the Wecken property.

Contribution

It introduces explicit solutions for orientable quadratic equations in free groups, including minimal solutions and their geometric interpretations.

Findings

Explicit solutions for quadratic equations in free groups are constructed.

Solutions depend on subgroup images under natural homomorphisms.

Geometric properties of associated surface maps vary with subgroup index.

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.