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Half-arc-transitive graphs are a fascinating topic which connects graphtheory, Riemann surfaces and group theory. Although fruitful results have beenobtained over the last half a century, it is still challenging to constructhalf-arc-transitive graphs with prescribed vertex stabilizers. Until recently,there have been only six known connected tetravalent half-arc-transitive graphswith nonabelian vertex stabilizers, and the question whether there exists aconnected tetravalent half-arc-transitive graph with nonabelian vertexstabilizer of order $2^s$ for every $s\geqslant3$ has been wide open. Thisquestion is answered in the affirmative in this paper via the construction of aconnected tetravalent half-arc-transitive graph with vertex stabilizer$\mathrm{D}_8^2\times\mathrm{C}_2^m$ for each integer $m\geqslant1$, where$\mathrm{D}_8^2$ is the direct product of two copies of the dihedral group oforder $8$ and $\mathrm{C}_2^m$ is the direct product of $m$ copies of thecyclic group of order $2$. The graphs constructed have surprisingly manysignificant properties in various contexts.

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