We initiate an investigation of lattices in a new class of locally compact groups: so-called locally pro- ##IMG## [http://ej.iop.org/images/1064-5616/211/8/1065/toc_MSB_211_8_1065ieqn1.gif] {$p$} -complete Kac-Moody groups. We discover that in rank 2 their cocompact lattices are particularly well- behaved: under mild assumptions, a cocompact lattice in this completion contains no elements of order ##IMG## [http://ej.iop.org/images/1064-5616/211/8/1065/toc_MSB_211_8_1065ieqn1.gif] {$p$} . This statement is still an open question for the Caprace-Rémy-Ronan completion. Using this, modulo results of Capdeboscq and Thomas, we classify edge-transitive cocompact lattices and describe a cocompact lattice of minimal covolume. Bibliography: 22 titles.

中文翻译：

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局部pro- ## IMG ## [http://ej.iop.org/images/1064-5616/211/8/1065/toc_MSB_211_8_1065ieqn1.gif] {$ p $}-完整等级2 Kac-喜怒无常的团体

我们开始研究一类新的局部紧凑组中的晶格：所谓的局部pro- ## IMG ## [http://ej.iop.org/images/1064-5616/211/8/1065/toc_MSB_211_8_1065ieqn1 .gif] {$ p $}-完整的Kac-Moody组。我们发现在第2级中，它们的共紧晶格表现得特别好：在温和的假设下，此完成中的共紧晶格不包含## IMG ##阶的元素[http://ej.iop.org/images/1064- 5616/211/8/1065 / toc_MSB_211_8_1065ieqn1.gif] {$ p $}。对于Caprace-Rémy-Ronan的完工，这一声明仍然是一个悬而未决的问题。使用Capdeboscq和Thomas的模结果，我们对边缘传递共紧凑格进行分类，并描述了最小体积的共紧凑格。参考书目：22种。