Fine graph

Farey graph

In graph theory , a branch of mathematics , fine graphs are a class of graphs with certain local finiteness properties. Fine graphs play a role in geometric group theory , particularly in the context of hyperbolicity and relative hyperbolicity of graphs and groups.

definition

A graph is called fine if it satisfies one (and therefore each) of the following equivalent conditions: ${\displaystyle K=(V,E)}$

• For each edge and each there are only finitely many circles of length running through . ${\displaystyle e\in E}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle e}$ ${\displaystyle n}$
• For all nodes and each there are only a finite number and connecting paths without repeating nodes. ${\displaystyle x,y\in V}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle x}$ ${\displaystyle y}$
• For all there is no infinite set and connecting pairwise independent paths without repeating nodes of length . (Here two paths are called independent if they only have the start and end point in common.) ${\displaystyle x,y\in V,n\in \mathbb {N} }$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle n}$
• If a pair of different vertices and is and and an edge-finite set of connected subgraphs of , which all have vertices and and contain, then must be finite. (Here a set is called finite if each edge is only contained in a finite number of subgraphs from .) ${\displaystyle x,y\in V}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle {\mathcal {L}}}$ ${\displaystyle K}$ ${\displaystyle n}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle {\mathcal {L}}}$ ${\displaystyle {\mathcal {L}}}$ ${\displaystyle e\in E}$ ${\displaystyle {\mathcal {L}}}$
• For every node the neighborhood is locally finite in . (That is, every node in is adjacent to only a finite number of nodes in .) ${\displaystyle x\in V}$ ${\displaystyle N_{K}(x)}$ ${\displaystyle K-\left\{x\right\}}$ ${\displaystyle N_{K}(x)}$ ${\displaystyle K-\left\{x\right\}}$ ${\displaystyle N_{K}(x)}$

Examples

• Locally finite graphs are fine.
• The Farey graph is fine.

literature

• Brian Bowditch: Relatively hyperbolic groups . (PDF) In: Internat. J. Algebra Comput. . 22, No. 3, 2012. doi : 10.1142 / S0218196712500166 . , Section 2