We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
EMBEDDING OF METRIC GRAPHS ON HYPERBOLIC SURFACES.
- Authors
SANKI, BIDYUT
- Abstract
An embedding of a metric graph $(G,d)$ on a closed hyperbolic surface is essential if each complementary region has a negative Euler characteristic. We show, by construction, that given any metric graph, its metric can be rescaled so that it admits an essential and isometric embedding on a closed hyperbolic surface. The essential genus $g_{e}(G)$ of $(G,d)$ is the lowest genus of a surface on which such an embedding is possible. We establish a formula to compute $g_{e}(G)$ and show that, for every integer $g\geq g_{e}(G)$ , there is an embedding of $(G,d)$ (possibly after a rescaling of $d$) on a surface of genus $g$. Next, we study minimal embeddings where each complementary region has Euler characteristic $-1$. The maximum essential genus $g_{e}^{\max }(G)$ of $(G,d)$ is the largest genus of a surface on which the graph is minimally embedded. We describe a method for an essential embedding of $(G,d)$ , where $g_{e}(G)$ and $g_{e}^{\max }(G)$ are realised.
- Subjects
METRIC geometry; HYPERBOLIC geometry; GRAPH theory; BETTI numbers; ISOMETRICS (Mathematics)
- Publication
Bulletin of the Australian Mathematical Society, 2019, Vol 99, Issue 3, p508
- ISSN
0004-9727
- Publication type
Article
- DOI
10.1017/S0004972719000145