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

The free regular $\star$-monoid of rank $r$ is the freest $r$-generated regular monoid ${\mathbf{F}}_r^{\star }$ in which every element $m$ has a distinguished pseudo-inverse $m^{\star }$ satisfying $m{m}^{\star }m=m$ and $(m^{\star }{)}^{\star }=m$. We study the growth rate of the monogenic regular $\star$-monoid ${\mathbf{F}}_1^{\star }$, and prove that this growth rate is intermediate. In particular, we deduce that ${\mathbf{F}}_r^{\star }$ is not rational or automatic for any $r\ge 1$, yielding the analogue of a result of Cutting & Solomon for free inverse monoids. Next, for all ranks $r\ge 1$ we determine the integral homology groups $H_{\ast }({\mathbf{F}}_r^{\star },\mathbb{Z})$, and by constructing a collapsing scheme prove that they vanish in dimension $3$ and above. As a corollary, we deduce that the free regular $\star$-monoid ${\mathbf{F}}_r^{\star }$ of rank $r\ge 1$ does not have the homological finiteness property ${\mathrm{FP}}_2$, yielding the analogue of a result of Gray & Steinberg for free inverse monoids.