Gap between LS-category and cohomological dimension for homomorphisms of geometrically finite groups

Characterize the possible values of cat(φ) − cdp(φ) for group homomorphisms between geometrically finite groups and ascertain whether arbitrarily large gaps can occur.

Background

The work investigates the relationship between LS-category cat(φ) and cohomological dimension cdp(φ) for group homomorphisms. Known examples show a gap of 1, but it is unclear whether larger gaps are possible, especially for geometrically finite groups.

A negative answer to the preceding inequality question (Question 6.11) would suggest constructions yielding arbitrarily large gaps between cat(φ) and cdp(φ), highlighting fundamental differences between these invariants.