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Commensurability invariants for nonuniform tree lattices

机译:非均匀树格的可通性不变量

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We study nonuniform lattices in the automorphism group G of a locally finite simplicial tree X. In particular, we are interested in classifying lattices up to commensurability in G. We introduce two new commensurability invariants: quotient growth, which measures the growth of the noncompact quotient of the lattice; and stabilizer growth, which measures the growth of the orders of finite stabilizers in a fundamental domain as a function of distance from a fixed basepoint. When X is the biregular tree X-m,X-n. we construct lattices realizing all triples of covolume, quotient growth, and stabilizer growth satisfying some mild conditions. In particular, for each positive real number nu we construct uncountably many noncommensurable lattices with covolume nu.
机译:我们研究局部有限的单纯树X的自同构群G中的非均匀格。特别是,我们对G中的格达到可共性的分类感兴趣。我们引入了两个新的可共性不变量:商增长,它衡量非紧商的增长晶格的稳定器增长,它测量基本稳定域中阶数稳定器的数量增长,该增长是与固定基点的距离的函数。当X是双正则树X-m,X-n。我们构建了满足某些温和条件的格子,以实现所有体积的三重,商增长和稳定剂增长。特别地,对于每个正实数nu,我们用无穷大的nu构造了无数不可数的晶格。

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