Elena Alexandrovna Kudryavtseva

Realization of primitive branched coverings over closed surfaces following the hurwitz approach

AbstractLet V be a closed surface, H⊑π1(V) a subgroup of finite index l and D=[A1,...,Am] a collection of partitions of a given number d≥2 with positive defect v(D). When does there exist a connected branched covering f:W→V of order d with branch data D and f∶W→Vn It has been shown by geometric arguments [4] that, for l=1 and a surface V different from the sphere and the projective plane, the corresponding branched covering exists (the data D is realizable) if and only if the data D fulfills the Hurwitz congruence v(D)э0 mod 2. In the case l>1, the corresponding branched covering exists if and only if v(D)э0 mod 2, the number d/l is an integer, and each partition Ai∈D splits into the union of l partitions of the number d/l. Here we give a purely algebraic proof of this result following the approach of Hurwitz [11].The realization problem for the projective plane and l=1 has been solved in [7,8]. The case of the sphere is treated in [1, 2, 12, 7].

Обобщение теоремы Бертрана на поверхности вращения@@@Generalization of Bertrand's theorem on surfaces of revolution

В работе доказано обобщение теоремы Бертрана на случай абстрактных поверхностей вращения, не имеющих “экваторов”. Доказан критерий существования на такой поверхности ровно двух центральных потенциалов (с точностью до аддитивной и мультипликативной констант), для которых все ограниченные орбиты замкнуты и имеется ограниченная неособая некруговая орбита. Доказан критерий существования ровно одного такого потенциала. Изучены геометрия и классификация соответствующих поверхностей, с указанием пары (гравитационного и осцилляторного) потенциалов или единственного (осцилляторного) потенциала. Показано, что на поверхностях, не относящихся ни к одному из описанных классов, потенциалов искомого вида не существует.

Realization of Primitive Branched Coverings Over Closed Surfaces

Let V be a closed surface, H ⊆ π1(V ) a subgroup of finite index and D = [A1, . . . , Am] a collection of partitions of a given number d ≥ 2 with positive defect v(D). When does there exist a connected branched covering f : W → V of order d with branch data D and f#(π1(W )) = H? We show that, for a surface V different from the sphere and the projective plane and = 1, the corresponding branched covering exists (the data D is realizable) if and only if the data D fulfills the Hurwitz congruence v(D) ≡ 0 mod 2. In the case > 1, the corresponding branched covering exists if and only if v(D) ≡ 0 mod 2, the number d/ is an integer, and each partition Ai ∈ D splits into the union of partitions of the number d/ . The realization problem for the projective plane and = 1 has been solved in (Edmonds-Kulkarni-Stong, 1984). The case of the sphere is treated in (BersteinEdmonds, 1979; Berstein-Edmonds, 1984; Husemoller, 1962; Edmonds-KulkarniStong, 1984). AMS: Primary: 55M20, Secondary: 57M12, 20F99

On multiplicity of mappings between surfaces

Let M and N be two closed (not necessarily orientable) surfaces, and f a continuous map from M to N. By definition, the minimal multiplicity MMR[f] of the map f denotes the minimal integer k having the following property: f can be deformed into a map g such that the number |g^{-1}(c)| of preimages of any point c in N under g is at most k. We calculate MMR[f] for any map

Some quadratic equations in the free group of rank 2

f

Roots of mappings on nonorientable surfaces and equations in free groups

of positive absolute degree A(f). The answer is formulated in terms of A(f), [pi_1(N):f_#(pi_1(M))], and the Euler characteristics of M and N. For a map f with A(f)=0, we prove the inequalities 2 <= MMR[f] <= 4.

Simple curves on surfaces and an analog of a theorem of Magnus for surface groups

For a given quadratic equation with any number of unknowns in any free group F , with right-hand side an arbitrary element of F , an algorithm for solving the problem of the existence of a solution was given by Culler [8] using a surface method and generalizing a result of Wicks [46]. Based on different techniques, the problem has been studied by the authors [11; 12] for parametric families of quadratic equations arising from continuous maps between closed surfaces, with certain conjugation factors as the parameters running through the group F . In particular, for a oneparameter family of quadratic equations in the free group F2 of rank 2, corresponding to maps of absolute degree 2 between closed surfaces of Euler characteristic 0, the problem of the existence of faithful solutions has been solved in terms of the value of the self-intersection index W F2! ZaF2c on the conjugation parameter. The present paper investigates the existence of faithful, or non-faithful, solutions of similar families of quadratic equations corresponding to maps of absolute degree 0. The existence results are proved by constructing solutions. The non-existence results are based on studying two equations in Zac and in its quotient Q, respectively, which are derived from the original equation and are easier to work with, where is the fundamental group of the target surface, and Q is the quotient of the abelian group Za nf1gc by the system of relations g g 1 , g2 nf1g. Unknown variables of the first and second derived equations belong to , Zac , Q, while the parameters of these equations are the projections of the conjugation parameter to and Q, respectively. In terms of these projections, sufficient conditions for the existence, or non-existence, of solutions of the quadratic equations in F2 are obtained.