Growth series of CAT(0) cubical complexes
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866916093544628224 |
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| author | Okun, Boris Scott, Richard |
| author_facet | Okun, Boris Scott, Richard |
| contents | Let $X$ be a CAT(0) cubical complex. The growth series of $X$ at $x$ is $G_{x}(t)=\sum_{y \in Vert(X)} t^{d(x,y)}$, where $d(x,y)$ denotes $\ell_{1}$-distance between $x$ and $y$. If $X$ is cocompact, then $G_{x}$ is a rational function of $t$. In the case when $X$ is the Davis complex of a right-angled Coxeter group it is a well-known that $G_{x}(t)=1/f_{L}(-t/(1+t))$, where $f_{L}$ denotes the $f$-polynomial of the link $L$ of a vertex of $X$. We obtain a similar formula for general cocompact $X$. We also obtain a simple relation between the growth series of individual orbits and the $f$-polynomials of various links. In particular, we get a simple proof of reciprocity of these series ($G_{x}(t)=\pm G_{x}(t^{-1})$) for an Eulerian manifold $X$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1812_07755 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Growth series of CAT(0) cubical complexes Okun, Boris Scott, Richard Group Theory Combinatorics 20F10, 20F55, 05A15 Let $X$ be a CAT(0) cubical complex. The growth series of $X$ at $x$ is $G_{x}(t)=\sum_{y \in Vert(X)} t^{d(x,y)}$, where $d(x,y)$ denotes $\ell_{1}$-distance between $x$ and $y$. If $X$ is cocompact, then $G_{x}$ is a rational function of $t$. In the case when $X$ is the Davis complex of a right-angled Coxeter group it is a well-known that $G_{x}(t)=1/f_{L}(-t/(1+t))$, where $f_{L}$ denotes the $f$-polynomial of the link $L$ of a vertex of $X$. We obtain a similar formula for general cocompact $X$. We also obtain a simple relation between the growth series of individual orbits and the $f$-polynomials of various links. In particular, we get a simple proof of reciprocity of these series ($G_{x}(t)=\pm G_{x}(t^{-1})$) for an Eulerian manifold $X$. |
| title | Growth series of CAT(0) cubical complexes |
| topic | Group Theory Combinatorics 20F10, 20F55, 05A15 |
| url | https://arxiv.org/abs/1812.07755 |