Explicit solutions of certain orientable quadratic equations in free groups

Autor(es)

Goncalves, D. ^[1] ; Nasybullov, T. ^[2]

Afiliação do(s) autor(es)

^[1] Univ Sao Paulo, Dept Math IME, Rua Matao 1010, BR-05508090 Butanta Sao Paulo, SP - Brazil

^[2] KU Leuven KULAK, Dept Math, Etienne Sabbelaan 53, B-8500 Kortrijk - Belgium

• Tipo de documento: Artigo Científico
• Fonte: INTERNATIONAL JOURNAL OF ALGEBRA AND COMPUTATION; v. 29, n. 8, p. 1451-1466, DEC 2019.
• Citações Web of Science: 0

Resumo

For g >= 1 denote by F-2g = < x(1), y(1), ..., x(g),y(g)> the free group on 2g generators and let B-g = {[}x(1), y(1)] ... {[}x(g), y(g)]. For l, c >= 1 and elements w(1), ..., w(l) is an element of F-2g, we study orientable quadratic equations of the form {[}u(1), v(1)] ... {[}u(h), v(h)] = (B-g(w1))(c)(B-g(w2))(c)...(B-g(wl))(c) with unknowns u(1), v(1), ..., 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, ..., 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 = < u(1), v(1), ..., u(h), v(h)> 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(x(1)),p(y(1)(l))>. We also provide an explicit solution of the equation {[}u(1), v(1)] ... {[}u(k), v(k)] = (B-1)(k+l)(B-1(y1))(k-l) for k > l >= 0 in F-2, and prove that if l not equal 0, then every solution of this equation is primitive. As a geometrical consequence, for every solution, we obtain a map f : S-h -> 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 vertical bar p(F-2) : p(H)vertical bar, such maps have fundamentally different geometric properties: in some cases, they satisfy the Wecken property and in other cases not. (AU)

• Processo FAPESP: 16/24707-4 - Topologia algébrica, geométrica e diferencial
• Beneficiário: Daciberg Lima Gonçalves
• Modalidade de apoio: Auxílio à Pesquisa - Temático