2018/08/25 by Daciberg Lima Gonçalves, Gonçalves, D. L., Timur Nasybullov +1
Mathematics · #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Geometry and complex manifolds #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.1808.08456
openalex publication_date 2018/08/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
For g≥1 denote by F2g=⟨ x1, y1,…,xg,yg⟩ the free group on 2g generators and by Bg=[x1,y1]…[xg,yg]. For l,c≥ 1 and elements w1,…,wl∈ F2g we study orientable quadratic equations of the form [u1,v1]…[uh,vh]=(Bgw1)c(Bgw2)c…(Bgwl)c with unknowns u1,v1,…,uh,vh and provide explicit solutions for them for the minimal possible number h. In the particular case when g=1, wi=y1i-1 for i=1,…,l and h the minimal number which satisfies h ≥ l(c-1)/2+1 we provide two types of solutions depending on the image of the subgroup H=⟨ u1,v1,…,uh,vh⟩ generated by the solution under the natural homomorphism p:F2→ F2/[F2,F2]: the first solution, which is called a primitive solution, satisfies p(H)=F2/[F2,F2], the second solution satisfies p(H) = ⟨ p(x1),p(y1l)⟩. We also provide an explicit solution of the equation [u1,v1]…[uk, vk] = (B1)k+l (B1y)k-l for k>l≥0 in F2, and prove that if l≠0, then every solution of this equation is primitive. As a geometrical consequence, for every solution we obtain a map f:Sh→ T from the orientable surface Sh of genus h to the torus T=S1 which has the minimal number of roots among all maps from the homotopy class of f. Depending on the number |p(F2):p(H)| such maps have fundamentally different geometric properties: in some cases they satisfy the Wecken property and in other cases not.