On the quantitative quasi-isometry problem: Transport of Poincar茅 inequalities and different types of quasi-isometric distortion growth

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• 作者：Vladimir Shchur¹ ; ^{vs3@sanger.co.uk" class="auth_mail" title="E-mail the corresponding author} ; ^{vlshchur@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：51F99
• 刊名：Journal of Functional Analysis
• 出版年：2015
• 出版时间：15 November 2015
• 年：2015
• 卷：269
• 期：10
• 页码：3147-3194
• 全文大小：694 K

文摘

We consider a quantitative form of the quasi-isometry problem. We discuss several arguments which lead us to a number of results and bounds of quasi-isometric distortion: comparison of volumes, connectivity, etc. Then we study the transport of Poincaré constants by quasi-isometries and we give sharp lower and upper bounds for the homotopy distortion growth for a certain class of hyperbolic metric spaces, a quotient of a Heintze group id="mmlsi1" class="mathmlsrc">ixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123615003699&_mathId=si1.gif&_user=111111111&_pii=S0022123615003699&_rdoc=1&_issn=00221236&md5=e1871332a5524dfddff425dd91f257ca" title="Click to view the MathML source">R鈰塕ⁿiner hidden">i1.gif" overflow="scroll">i mathvariant="double-struck">Ri>鈰?/mo>i mathvariant="double-struck">Ri>i>ni> by id="mmlsi2" class="mathmlsrc">ixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123615003699&_mathId=si2.gif&_user=111111111&_pii=S0022123615003699&_rdoc=1&_issn=00221236&md5=4d067fc699857295a84b05e41bbed668" title="Click to view the MathML source">Zⁿiner hidden">i2.gif" overflow="scroll">i mathvariant="double-struck">Zi>i>ni>. We also prove the linear distortion growth between hyperbolic space id="mmlsi3" class="mathmlsrc">ixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123615003699&_mathId=si3.gif&_user=111111111&_pii=S0022123615003699&_rdoc=1&_issn=00221236&md5=4318951dfc903b88c1c20b240932d983" title="Click to view the MathML source">Hⁿ,n≥3iner hidden">i3.gif" overflow="scroll">i mathvariant="double-struck">Hi>i>ni>,i>ni>≥3 and a tree.