Tim Clausen: Mock hyperbolic reflection spaces and Frobenius groups of finite Morley rank
Thursday, 22.04.2021 10:30 im Raum via Zoom
Joint work with Katrin Tent.
A Frobenius group is a group G together with a proper nontrivial malnormal
subgroup H. A classical result due to Frobenius states that finite Frobenius
groups split, i.e. they can be written as a semidirect product of a normal
subgroup and the subgroup H. It is an open question if this holds true for
groups of finite Morley rank, and the existence of a non-split Frobenius group
of finite Morley rank would contradict the Algebraicity Conjecture. We use
mock hyperbolic reflection spaces, a generalization of real hyperbolic spaces,
to study Frobenius groups of finite Morley rank.
We show that the involutions in a connected Frobenius group of finite Morley
rank and odd type form a mock hyperbolic reflection space. These spaces
satisfy certain rank inequalities and we conclude that connected Frobenius
groups of odd type and Morley rank at most 6 split.
Moreover, by using a construction from the theory of K-loops we show that if G
is a connected Frobenius group of degenerate type with abelian complement,
then G can be expanded to a group whose involutiuons almost form a mock
hyperbolic reflection space. The rank inequalities allow us to show structural
results for such groups. As a special case we get Frecon's theorem: There is
no bad group of Morley rank 3.
Angelegt am Thursday, 15.04.2021 14:17 von Martina Pfeifer
Geändert am Thursday, 15.04.2021 14:17 von Martina Pfeifer
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