Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userCayley Graphs on Billiard Surfaces
155
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this article we discuss a connection between two famous constructions in mathematics: a Cayley graph of a group and a (rational) billiard surface. For each rational billiard surface, there is a natural way to draw a Cayley graph of a dihedral group on that surface. Both of these objects have the concept of genus attached to them. For the Cayley graph, the genus is defined to be the lowest genus amongst all surfaces that the graph can be drawn on without edge crossings. We prove that the genus of the Cayley graph associated to a billiard surface arising from a triangular billiard table is always zero or one. One reason this is interesting is that there exist triangular billiard surfaces of arbitrarily high genus , so the genus of the associated graph is usually much lower than the genus of the billiard surface.
rate research
Read More
We identify all translation covers among triangular billiard surfaces. Our main tools are the holonomy field of Kenyon and Smillie and a geometric property of translation surfaces, which we call the fingerprint of a point, that is preserved under balanced translation covers.
The genus graphs have been studied by many authors, but just a few results concerning in special cases: Planar, Toroidal, Complete, Bipartite and Cartesian Product of Bipartite. We present here a derive general lower bound for the genus of a abelian Cayley graph and construct a family of circulant graphs which reach this bound.
Let $G$ be a group and $Ssubseteq G$ its subset such that $S=S^{-1}$, where $S^{-1}={s^{-1}mid sin S}$. Then {it the Cayley graph ${rm Cay}(G,S)$} is an undirected graph $Gamma$ with the vertex set $V(Gamma)=G$ and the edge set $E(Gamma)={(g,gs)mid gin G, sin S}$. A graph $Gamma$ is said to be {it integral} if every eigenvalue of the adjacency matrix of $Gamma$ is integer. In the paper, we prove the following theorem: {it if a subset $S=S^{-1}$ of $G$ is normal and $sin SRightarrow s^kin S$ for every $kin mathbb{Z}$ such that $(k,|s|)=1$, then ${rm Cay}(G,S)$ is integral.} In particular, {it if $Ssubseteq G$ is a normal set of involutions, then ${rm Cay}(G,S)$ is integral.} We also use the theorem to prove that {it if $G=A_n$ and $S={(12i)^{pm1}mid i=3,dots,n}$, then ${rm Cay}(G,S)$ is integral.} Thus, we give positive solutions for both problems 19.50(a) and 19.50(b) in Kourovka Notebook.
We associate to triangulations of infinite type surface a type of flip graph where simultaneous flips are allowed. Our main focus is on understanding exactly when two triangulations can be related by a sequence of flips. A consequence of our results is that flip graphs for infinite type surfaces have uncountably many connected components.
Let $G$ be a finitely generated group acting faithfully and properly discontinuously by homeomorphisms on a planar surface $X subseteq mathbb{S}^2$. We prove that $G$ admits such an action that is in addition co-compact, provided we can replace $X$ by another surface $Y subseteq mathbb{S}^2$. We also prove that if a group $H$ has a finitely generated Cayley (multi-)graph $C$ covariantly embeddable in $mathbb{S}^2$, then $C$ can be chosen so as to have no infinite path on the boundary of a face. The proofs of these facts are intertwined, and the classes of groups they define coincide. In the orientation-preserving case they are exactly the (isomorphism types of) finitely generated Kleinian function groups. We construct a finitely generated planar Cayley graph whose group is not in this class. In passing, we observe that the Freudenthal compactification of every planar surface is homeomorphic to the sphere.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
