**Speaker:** Kay Magaard (Birmingham)

**Title:** * The Lift Invariant and the Conway-Parker Theorem*

**Abstract:**

It is a well known fact that the braid orbits on Nielsen tuples are in one to one correspondence with the connected components of Hurwitz spaces corresponding to curve covers of the Riemann sphere. The theorem of Conway and Parker asserts that if every element of the Schur multiplier of a group G is a commutator and if the ramification type is sufficiently general, then the corresponding Hurwitz space is connected. For many reasons it would be desirable to have effective versions of this theorem. In 2010 Fried showed that the lift invariant distinguishes components of Hurwitz spaces corresponding to alternating group covers whose ramification data consists entirely of three cycles. In particular the number of connected components of the corresponding Hurwitz space is never more than 2. In joint work with A. James and S.Shpectorov we show that the lift invariant distinguishes Hurwitz components of \(A_5\) covers. Generalizations of this will also be discussed.