Orbits of braid groups on cacti. (English) Zbl 1008.20030

Jones, Gareth A.; Zvonkin, Alexander

Mosc. Math. J. 2, No. 1, 127-160 (2002).

A complex polynomial \(p(z)\) of degree \(n\) is regarded as a branched \(n\)-sheeted cover of the sphere \(S^2\). If \(w_1,\dots,w_k\) are the critical values of \(p\), then a rotation of \(w=p(z)\) around \(w_i\) induces a ‘monodromy permutation’ \(g_i\) of the \(n\) sheets \(z=p^{-1}(w)\). These \(k\) monodromy permutations generate the ‘monodromy group’ \(G\) of \(p\). This group can be regarded as a subgroup of the symmetric group \(S_n\) consisting of the permutations of the sheets obtained by lifting closed paths in \(\mathbb{C}\setminus\{w_1,\dots,w_k\}\). Certain properties of the group \(G\) reflect properties of the polynomial \(p\). For \(i=1,\dots,k\), let \(n_i\) be the number of cycles of \(g_i\) (which is equal to the cardinality of \(p^{-1}(w_i)\)). The Riemann-Hurwitz formula for \(n\)-sheeted covers implies the following ‘polarity condition’: \[ \sum_{i=1}^kn_i=(k-1)n+1. \] A ‘cactus’ is an ordered \(k\)-tuple of elements \(g_1,\dots,g_k\) of \(S_n\) satisfying the polarity condition, and such that the product \(g_1\cdots g_k\) is the cycle \((0,1,\dots,n-1)\). Cacti have been used in the topological classification of polynomials, where two polynomials \(p\) and \(q\) are said to be ‘topologically equivalent’ if \(q\circ h_1=h_2\circ p\) for some orientation-preserving self-homeomorphisms \(h_1\) and \(h_2\) of \(S^2\). The classification of polynomials up to topological equivalence is a basic subject of research which started in the 19th century.
In this paper, the authors use methods from enumerative combinatorics, group theory and character theory to study cacti. They use computational techniques to split cacti into braid group orbits. They give several examples of such splittings, and they use monodromy groups to explain the different orbits. The examples are motivated by a theorem of P. Müller [Contemp. Math. 186, 385-401 (1995; Zbl 0840.12001)]. It is known that polynomials with more than two finite critical values have exactly three exceptional monodromy groups (exceptional meaning here not equal to \(S_n\) or to \(A_n\), where \(n\) is the degree), one in degree 7, one in degree 13 and one in degree 15. Using the cacti techniques, the authors give a complete topological classification of these three exceptional cases.

Reviewer: Athanase Papadopoulos (Strasbourg)

MSC:

20F36	Braid groups; Artin groups
57M12	Low-dimensional topology of special (e.g., branched) coverings
30F10	Compact Riemann surfaces and uniformization
57M60	Group actions on manifolds and cell complexes in low dimensions
20B15	Primitive groups
05C25	Graphs and abstract algebra (groups, rings, fields, etc.)
12D05	Polynomials in real and complex fields: factorization

Keywords:

topological classification of complex polynomials; monodromoy groups; braid group actions; cacti; branched cover of the sphere; critical values

Citations:

Zbl 0840.12001