Tetravalent half-arc-transitive graphs with unbounded nonabelian vertex stabilizers

Abstract

Half-arc-transitive graphs are a fascinating topic which connects graph theory, Riemann surfaces and group theory. Although fruitful results have been obtained over the last half a century, it is still challenging to construct half-arc-transitive graphs with prescribed vertex stabilizers. Until recently, there have been only six known connected tetravalent half-arc-transitive graphs with nonabelian vertex stabilizers, and the question whether there exists a connected tetravalent half-arc-transitive graph with nonabelian vertex stabilizer of order $2^s$ for every $s⩾3$ has been wide open. This question is answered in the affirmative in this paper via the construction of a connected tetravalent half-arc-transitive graph with vertex stabilizer ${\mathrm{D}}_8^2\times {\mathrm{C}}_2^m$ for each integer $m⩾1$, where ${\mathrm{D}}_8^2$ is the direct product of two copies of the dihedral group of order $8$ and ${\mathrm{C}}_2^m$ is the direct product of $m$ copies of the cyclic group of order $2$. The graphs constructed have surprisingly many significant properties in various contexts.