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EMBEDDING OF METRIC GRAPHS ON HYPERBOLIC SURFACES
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 99 / Issue 3 / June 2019
- Published online by Cambridge University Press:
- 13 February 2019, pp. 508-520
- Print publication:
- June 2019
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- Article
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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.
