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

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 century, it is still challenging to construct half-arc-transitive with prescribed vertex stabilizers. Until recently, there only six known connected tetravalent nonabelian stabilizers, question whether exists stabilizer of order 2s for every s⩾3 has wide open. This answered in affirmative this paper via construction D82×C2m each integer m⩾1, where D82 direct product two copies dihedral 8 C2m m cyclic 2. The constructed surprisingly many significant properties various contexts.