Read more about the mathDefinitions, the finiteness proof and worked examples, with figures you can move.
Give every edge of a graph a positive length. As lengths shrink to zero, edges contract and the graph degenerates. The moduli space Mg,n collects all such metric graphs of genus g with n marked legs. It has one cell ℝ>0E(G)/Aut(G) for each stable graph G, and the cells are glued along edge contractions.
Pick a type (g, n) below to see its stable graphs, the poset they form under contraction, their automorphisms, and, when the dimension is small, a picture of the space you can move around in.
Click an edge in the drawing to contract it. Click an automorphism to watch where it sends each half-edge.