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| Format: | Preprint |
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2018
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| Online Access: | https://arxiv.org/abs/1807.08827 |
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| _version_ | 1866911910145818624 |
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| author | Zamora, Sergio |
| author_facet | Zamora, Sergio |
| contents | We study the geometry of compact geodesic spaces with trivial first Betti number admitting large finite groups of isometries. We show that if a finite group $G$ acts by isometries on a compact geodesic space $X$ whose first Betti number vanishes, then diam$(X) / $diam$(X / G ) \leq 4 \sqrt{ \vert G \vert }$.
For a group $G$ and a finite symmetric generating set $S$, $P_k(Γ(G, S))$ denotes the 2-dimensional CW-complex whose 1-skeleton is the Cayley graph $Γ$ of $G$ with respect to $S$ and whose 2-cells are $m$-gons for $0 \leq m \leq k$, defined by the simple graph loops of length $m$ in $Γ$, up to cyclic permutations.
Let $G$ be a finite abelian group with $\vert G \vert \geq 3$ and $S$ a symmetric set of generators for which $P_k(Γ(G,S))$ has trivial first Betti number. We show that the first nontrivial eigenvalue $-λ_1$ of the Laplacian on the Cayley graph satisfies $λ_1 \geq 2 - 2 \cos ( 2 π/ k ) $. We also give an explicit upper bound on the diameter of the Cayley graph of $G$ with respect to $S$ of the form $O (k^2 \vert S \vert \log \vert G \vert )$. Related explicit bounds for the Cheeger constant and Kazhdan constant of the pair $(G,S)$ are also obtained. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1807_08827 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Fundamental groups and group presentations with bounded relator lengths Zamora, Sergio Metric Geometry Algebraic Topology Group Theory 51Fxx, 05E16, 20B25 We study the geometry of compact geodesic spaces with trivial first Betti number admitting large finite groups of isometries. We show that if a finite group $G$ acts by isometries on a compact geodesic space $X$ whose first Betti number vanishes, then diam$(X) / $diam$(X / G ) \leq 4 \sqrt{ \vert G \vert }$. For a group $G$ and a finite symmetric generating set $S$, $P_k(Γ(G, S))$ denotes the 2-dimensional CW-complex whose 1-skeleton is the Cayley graph $Γ$ of $G$ with respect to $S$ and whose 2-cells are $m$-gons for $0 \leq m \leq k$, defined by the simple graph loops of length $m$ in $Γ$, up to cyclic permutations. Let $G$ be a finite abelian group with $\vert G \vert \geq 3$ and $S$ a symmetric set of generators for which $P_k(Γ(G,S))$ has trivial first Betti number. We show that the first nontrivial eigenvalue $-λ_1$ of the Laplacian on the Cayley graph satisfies $λ_1 \geq 2 - 2 \cos ( 2 π/ k ) $. We also give an explicit upper bound on the diameter of the Cayley graph of $G$ with respect to $S$ of the form $O (k^2 \vert S \vert \log \vert G \vert )$. Related explicit bounds for the Cheeger constant and Kazhdan constant of the pair $(G,S)$ are also obtained. |
| title | Fundamental groups and group presentations with bounded relator lengths |
| topic | Metric Geometry Algebraic Topology Group Theory 51Fxx, 05E16, 20B25 |
| url | https://arxiv.org/abs/1807.08827 |