تقرير
On the growth and integral (co)homology of free regular star-monoids
| العنوان: | On the growth and integral (co)homology of free regular star-monoids |
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| المؤلفون: | Nyberg-Brodda, Carl-Fredrik |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Group Theory, Mathematics - Rings and Algebras |
| الوصف: | 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 $mm^\star m = m$ 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 $r \geq 1$, yielding the analogue of a result of Cutting & Solomon for free inverse monoids. Next, for all ranks $r \geq 1$ 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 $r \geq 1$ does not have the homological finiteness property $\operatorname{FP}_2$, yielding the analogue of a result of Gray & Steinberg for free inverse monoids. Comment: 15 pages. Comments welcome! |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2408.05986 |
| رقم الأكسشن: | edsarx.2408.05986 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |