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| Thompson's Group $F$ and the Linear Group GL$_{\infty}(\mathbb{Z})$\ |
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Citation： |
Yan WU,Xiaoman CHEN.Thompson's Group $F$ and the Linear Group GL$_{\infty}(\mathbb{Z})$\[J].Chinese Annals of Mathematics B,2011,32(6):863~884 |
| Page view： 1755
Net amount： 1150 |
Authors： |
Yan WU; Xiaoman CHEN; |
Foundation： |
the National Natural Science Foundation of China (No. 10731020) and the Shanghai
Natural Science Foundation of China (No. 09ZR1402000). |
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| Abstract： |
The authors study the finite decomposition complexity of metric
spaces of $H$, equipped with different metrics, where $H$ is a
subgroup of the linear group ${\rm GL}_{\infty}(\mathbb{Z})$. It is
proved that there is an injective Lipschitz map $\varphi:(F,
d_{S})\rightarrow(H,d)$, where $F$ is the Thompson's group, $d_{S}$
the word-metric of $F$ with respect to the finite generating set $S$
and $d$ a metric of $H$. But it is not a proper map. Meanwhile, it
is proved that $\varphi:(F, d_{S})\rightarrow(H,d_{1})$ is not a
Lipschitz map, where $d_{1}$ is another metric of $H$. |
Keywords： |
Finite decomposition complexity, Thompson’s group F, Word-metric,
Lipschitz map, Reduced tree diagram |
Classification： |
46L07, 46L80 |
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