Quasi-isometries between non-locally-finite graphs and structure trees

Tree amalgamations and quasi-isometries

We investigate the connections between tree amalgamations and quasi-isometries. In particular, we prove that the quasi-isometry type of multi-ended accessible quasi-transitive connected locally finite graphs is determined by the quasi-isometry type of their one-ended factors in any of their terminal factorisations. Our results carry over theorems of Papsoglu and Whyte on quasi-isometries between multi-ended groups to those between multi-ended graphs. In the end, we discuss the impact of our results to a question of Woess.

可及性、平面图与准等距性

Accessibility, planar graphs, and quasi-isometries

本文证明了可及的平面图与准等距图之间的关系，揭示了有限生成群在准等距下的分类，指出此类群本质上是自由积的虚拟自由群或曲面群。

拟等距、收缩与交图

Quasi-isometries, contractions, and intersection graphs

本文证明图为拟平面图当且仅经两类有界迭代操作得到，推广了字符串图为拟平面图的结论，还推导了拟平面图的收缩子式性质及树分解相关结果，为拟等距等提供了工具。

具有可变边长的图之间的拟等距映射

Quasi-isometries between graphs with variable edge lengths

本文研究可变边长图之间的拟等距映射，并否定了 Nguyen、Scott 和 Seymour 关于通过调整目标图边长和加性畸变常数即可消除乘性畸变因子的猜想。

加法拟等距与仙人掌图

Additive quasi-isometries and cacti

本文证明不含胖θ曲线的测地度量空间与仙人掌图存在加法拟等距，并由此推出所有拟等距于仙人掌图的空间亦然。

胖子式无法（通过拟等距）变薄

Fat minors cannot be thinned (by quasi-isometries)

本研究推翻了Georgakopoulos与Papasoglu关于含K-胖H子式的长度空间/图与无H子式的图拟等距的猜想，给出反例并证明了不含K-胖H子式的图与不含3-胖H子式的图拟等距的弱化结论。