Leighton's Theorem: extensions, limitations, and quasitrees

Abstract

Leighton's Theorem states that if there is a tree $T$ that covers two finite graphs $G_1$ and $G_2$, then there is a finite graph $\hat{G}$ that is covered by $T$ and covers both $G_1$ and $G_2$. We prove that this result does not extend to regular covers by graphs other than trees. Nor does it extend to non-regular covers by a quasitree, even if the automorphism group of the quasitree contains a uniform lattice. But it does extend to regular coverings by quasitrees.

29 pages, 10 figures; v2: minor changes made following referee's comments; v3: minor edits to proof of Theorem 1.1 and correction of Example 2.2; to appear in Algebraic & Geometric Topology

References in corpus (2)

• Poc Sets, Median Algebras and Group Actions
• Quasi-isometric rigidity for graphs of virtually free groups with two-ended edge groups

Cited by in corpus (2)

• Small non-Leighton two-complexes
• Tiny non-Leighton complexes