Summary: SIMPLICIAL EMBEDDINGS BETWEEN PANTS GRAPHS
Javier Aramayona 1
We prove that, except in some low-complexity cases, every locally in-
jective simplicial map between pants graphs is induced by a 1-injective
embedding between the corresponding surfaces.
1 Introduction and main results
To a surface one may associate a number of naturally defined objects
its Teichm¨uller space, mapping class group, curve or pants graph, etc.
An obvious problem is then to study embeddings between objects in the
same category, where the term "embedding" is to be interpreted suitably
in each case, for instance "isometric embedding" in the case of Teichm¨uller
spaces, "injective homomorphism" in the case of mapping class groups, and
"injective simplicial map" in the case of curve and pants graphs.
For pants graphs, this problem was first studied by D. Margalit [Mar],
who showed that every automorphism of the pants graph is induced by a
surface homeomorphism. More concretely, let be a compact orientable
surface whose every connected component has negative Euler characteristic.
The pants graph P() of is a simplicial graph whose vertices are pants de-
compositions of and whose edges correspond to elementary moves on pants