Uniform tree

In mathematics, a uniform tree is a locally finite tree which is the universal cover of a finite graph. Equivalently, the full automorphism group ${\displaystyle G=Aut(X)}$ of the tree, which is a locally compact topological group, is unimodular and ${\displaystyle G/X}$ is finite. Also equivalent is the existence of a uniform X-lattice in ${\displaystyle G}$.

For a graph ${\displaystyle G}$ which contains no cycles, ${\displaystyle G}$ is its own uniform tree. If ${\displaystyle G}$ contains at least 1 cycle, its uniform tree is an infinite tree.

Leighton's Graph Covering Theorem states that any two finite graphs that share a common covering must also share a common finite covering. Walter D. Neumann expanded on this in 2011, proving any two graphs that have a common covering necessarily have the same universal covering. This means that every uniform tree corresponds to a unique family of finite graphs.

• Covering graph

• Bass, Hyman; Lubotzky, Alexander (2001), Tree Lattices, Progress in Mathematics, vol. 176, Birkhäuser, ISBN 0-8176-4120-3
• Neumann, Walter D. (2011). "On Leighton's graph covering theorem". Groups, Geometry, and Dynamics. 4: 863–872. arXiv:0906.2496.

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