Combinatorial Theory

We study large uniform random maps with one face whose genus grows linearly with the number of edges. They can be seen as a model of discrete hyperbolic geometry. In the past, several of these hyperbolic geometric features have been discovered, such as their local limit or their logarithmic diameter. In this work, we show that with high probability such a map contains a very large induced subgraph that is an expander.

Mathematics Subject Classifications: 05C10, 05C48, 60C05, 60D05

Keywords: Combinatorial maps, high genus, expander graphs