On tetravalent half-arc-transitive graphs

Vertex-stabilizers of trivalent edge-transitive graphs have been classified by Tutte, Goldschmidt and some others in several previous papers. Tetravalent half-arc-transitive graphs form an important class of tetravalent edge-transitive graphs. Maru\v{s}i\v{c} and Nedela (2001) initiated the study of the problem of classifying vertex-stabilizers of tetravalent half-arc-transitive graphs, which has received extensive attention and considerable effort in the literature. In this paper, we solve this problem by proving that a group is the vertex-stabilizer of a connected tetravalent half-arc-transitive graph if and only if it is a non-trivial concentric group. Note that a characterization of concentric groups has been given by Maru\v{s}i\v{c} and Nedela in 2001. Furthermore, we give an explicit construction of an infinite family of tetravalent half-arc-transitive graphs with automorphism group isomorphic to $A_{2^n}\wr \mathbb{Z}_2$ and vertex-stabilizers isomorphic to $(D_8^2\times\mathbb{Z}_{2}^{n-6})^2$ for $n\geq7$. These are the first known family of basic tetravalent half-arc-transitive graphs of bi-quasiprimitive type.