Filling of closed surfaces

Abstract

Let Fg denote a closed oriented surface of genus g. A set of simple closed curves is called a filling of Fg if its complement is a disjoint union of discs. The mapping class group Mod(Fg) of genus g acts on the set of fillings of Fg. The union of the curves in a filling forms a graph on the surface which is a so-called decorated fat graph. It is a fact that two fillings of Fg are in the same Mod(Fg)-orbit if and only if the corresponding fat graphs are isomorphic. We prove that any filling of F 2 whose complement is a single disc (i.e., a so-called minimal filling) has either three or four closed curves and in each of these two cases, there is a unique such filling up to the action of Mod(F 2 ).We provide a constructive proof to show that the minimum number of discs in the complement of a filling pair of F 2 is two. Finally, given positive integers g and k with (g, k) = (2, 1), we construct a filling pair of Fg such that the complement is a union of k topological discs.

Systolic Fillings of Surfaces

“…A cycle in a decorated fat graph is called a standard cycle if every two consecutive edges in the cycle are opposite each other with respect to the cyclic order on the set of edges emanating from their common vertex. For more details on fat graphs, we refer the reader to [7,Section 2].…”

“…Note that, for a formal definition of a fat graph and examples with such descriptions of fat graphs, we refer the reader to [7,Section 2].…”

Geometric realizations of cyclic actions on surfaces

“…A fat graph [13] is a graph equipped with a cyclic order on the set of edges incident at each vertex. If the valency of a vertex is less than three, then the cyclic order on the set of edges incident at that vertex is trivial.…”

“…A decorated fat graph can be written as an edge-disjoint union of standard cycles. Let Σ(G) be the surface with boundary associated with G. Then the number of boundary components in Σ(G) is the same as the number of orbits of σ 1 σ −1 0 (see [13] for more details on this topic).…”

“…In Section 3, we give a combinatorial perspective to these realizations by establishing a correspondence between certain finite order mapping classes and fat graph automorphisms [13]. Let Γ be fat graph with one boundary component with a vertex set V and an edge set E. An automorphism F of Γ of order n is said to be irreducible if with either |E| = n and |V / F | = 1 or |E| = 2n and |V / F | = 2.…”