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Commensurability invariants for nonuniform tree lattices

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Abstract

We study nonuniform lattices in the automorphism groupG of a locally finite simplicial treeX. In particular, we are interested in classifying lattices up to commensurability inG. We introduce two new commensurability invariants:quotient growth, which measures the growth of the noncompact quotient of the lattice; andstabilizer growth, which measures the growth of the orders of finite stabilizers in a fundamental domain as a function of distance from a fixed basepoint. WhenX is the biregular treeX m,n, we construct lattices realizing all triples of covolume, quotient growth, and stabilizer growth satisfying some mild conditions. In particular, for each positive real numberν we construct uncountably many noncommensurable lattices with covolumeν.

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Authors and Affiliations

  1. Department of Mathematics, University of Chicago, 5734 S. University Ave, 60637, Chicago, IL, USA

    Benson Farb & G. Christopher Hruska

Authors
  1. Benson Farb
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  2. G. Christopher Hruska
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Correspondence to Benson Farb.

Additional information

Supported in part by NSF grants DMS-9704640 and DMS-0244542.

Supported in part by an NSF postdoctoral research fellowship.

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Farb, B., Hruska, G.C. Commensurability invariants for nonuniform tree lattices. Isr. J. Math. 152, 125–142 (2006). https://doi.org/10.1007/BF02771979

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  • DOI: https://doi.org/10.1007/BF02771979

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