Quasi-isometries between groups with infinitely many ends
Quasi-isometries between groups with infinitely many ends
复制标题
具有无限多端的群之间的拟等距
DOI：
10.1007/s00014-002-8334-2
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发表时间：
2002
影响因子：
0.9
通讯作者：
Kevin Whyte
中科院分区：
文献类型：
--
作者：
Panos Papazoglu;Kevin Whyte
抽象的。设G，F是具有无穷多个端点的有限生成群，且
$\pi_1(\Gamma，\数学A)，\pi_1(\Delta，\数学B)$是F，G的群分解图，使得所有的边群都是有限的，并且所有的顶点群至多有一端。我们证明了G，F是拟等距的当且仅当
$\pi_1(\Gamma，\Mathcal A)$拟等距于
$\pi_1(\Delta，\Mathcal B)$和
$\pi_1(\Delta，\Mathcal B)$拟等距于？的某个单端顶点群
$\pi_1(\Gamma，\数学A)$。从我们的证明还可以得出，如果G是任何至少三阶的有限生成群，则群：
$G\ast G，G\ast\mathbb{Z}，G\ast G\ast G$and
$G\ast\mathbb{Z}/2\mathbb{Z}$都是拟等距的。
Abstract. Let G, F be finitely generated groups with infinitely many ends and let¶
$ \pi_1(\Gamma,\mathcal A), \pi_1(\Delta ,\mathcal B) $ be graph of groups decompositions of F, G such that all edge groups are finite and all vertex groups have at most one end. We show that G, F are quasi-isometric if and only if every one-ended vertex group of
$ \pi_1(\Gamma, \mathcal A) $ is quasi-isometric to some one-ended vertex group of
$ \pi_1(\Delta, \mathcal B) $ and every one-ended vertex group of
$ \pi_1(\Delta, \mathcal B) $ is quasi-isometric to some one-ended vertex group of¶
$ \pi_1(\Gamma, \mathcal A) $. From our proof it also follows that if G is any finitely generated group, of order at least three, the groups:
$ G \ast G, G \ast \mathbb{Z}, G \ast G \ast G $ and
$ G \ast \mathbb{Z}/2\mathbb{Z} $ are all quasi-isometric.