On the growth and integral (co)homology of free regular star-monoids

The free regular -monoid of rank r is the freest r-generated regular monoid Fᵣ^ in which every element m has a distinguished pseudo-inverse m^ satisfying mm^ m = m and (m⁾^ = m. We study the growth rate of the monogenic regular -monoid F₁^, and prove that this growth rate is intermediate. In particular, we deduce that Fᵣ^ is not rational or automatic for any r ≥ 1, yielding the analogue of a result of Cutting & Solomon for free inverse monoids. Next, for all ranks r ≥ 1 we determine the integral homology groups H₍Fᵣ^, Z), and by constructing a collapsing scheme prove that they vanish in dimension $3$ and above. As a corollary, we deduce that the free regular -monoid Fᵣ^ of rank r ≥ 1 does not have the homological finiteness property FP₂, yielding the analogue of a result of Gray & Steinberg for free inverse monoids.