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.

References

Question 6.12. What is the gap of LS-category and cohomological dimension of group homo-

morphisms of geometrically finite groups? All known examples regarding the Question 6.12 have a gap of 1 (See [DK, DD, Gr]). Possible negative answer for Question 6.11 comes the constructing the arbitrary large gap for Question 6.12.

— On the sequential topological complexity of group homomorphisms  (2402.13389 - Kuanyshov, 2024) in Question 6.12 (Section 6), page 18

At this point, we would like to mention a question whether there is an upper estimate of (d'\ast d) in terms of (d) and (d'), perhaps by their sum. In terms of our notation in Proposition \ref{prop:secat of composition}, Arkowitz and Strom Theorem 5.4 proved that (s\circ s')\le {FH}(s)+(s')$, where ${FH}(s)$ denotes a version of category of a map introduced by Fadell and Husseini (the latter can be also expressed in terms of the relative category of the pair $(E,A)$, see Section 7.2). Unfortunately, we do not know the precise relation between ${FH}(\pi{d'})$ and (d').

— Parametrized LS category and group actions  (2609.01462 - Ogrinec et al., 1 Sep 2026) in Section 3, subsection “Compositions of splittings”