Let Mod(Sg) denote the mapping class group of the closed orientable surface Sg of genus g ≥ 2, and let f ∈ Mod(Sg) be of finite order. We give an inductive procedure to construct an explicit hyperbolic structure on Sg that realizes f as an isometry. In other words, this procedure yields an explicit solution to the Nielsen realization problem for cyclic subgroups of Mod(Sg). Furthermore, we give a purely combinatorial perspective by showing how certain finite order mapping classes can be viewed as fat graph automorphisms. As an application of our realizations, we determine the sizes of maximal reduction systems for certain finite order mapping classes. Moreover, we describe a method to compute the image of finite order mapping classes and the roots of Dehn twists, under the symplectic representation Ψ : Mod(Sg) → Sp(2g; Z).2000 Mathematics Subject Classification. Primary 57M60; Secondary 57M50, 57M99.

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.

A filling of a closed hyperbolic surface is a set of simple closed geodesics whose complement is a disjoint union of hyperbolic polygons. The systolic length is the length of a shortest essential closed geodesic on the surface. A geodesic is called systolic, if the systolic length is realised by its length. For every $g\geq 2$, we construct closed hyperbolic surfaces of genus $g$ whose systolic geodesics fill the surfaces with complements consisting of only two components. Finally, we remark that one can deform the surfaces obtained to increase the systole.

Abstract. Given a hyperbolic surface, the set of all closed geodesics whose length is minimal form a graph on the surface, in fact a so-called fat graph, which we call the systolic graph. We study which fat graphs are systolic graphs for some surface (we call these admissible).There is a natural necessary condition on such graphs, which we call combinatorial admissibility. Our first main result is that this condition is also sufficient.It follows that a sub-graph of an admissible graph is admissible. Our second major result is that there are infinitely many minimal non-admissible fat graphs (in contrast, for instance, to the classical result that there are only two minimal non-planar graphs).

The isoperimetric inequality in the Euclidean geometry (for polygons) states that among all n-gons having a fixed perimeter p, the one with the largest area is the regular n-gon. One can generalise this result to simple closed curves; in this case, the curve with the maximum area is the circle. The statement is true in hyperbolic geometry as well (see Bezdek [2]).In this paper, we generalize the isoperimetric inequality to disconnected regions, i.e. we allow the area to be split between regions. We give necessary and sufficient conditions for the isoperimetric inequality (in Euclidean and hyperbolic geometry) to hold for multiple n-gons whose perimeters add up to p.