Arbitrary Real Volumes, Cusps, and Homology

  • Hyman Bass
  • Alexander Lubotzky
Part of the Progress in Mathematics book series (PM, volume 176)


Let X be a locally finite tree and Γ a non-uniform-lattice. In this section we show that, even when X is regular (of degree ≥ 3), Vol(Γ\\X) can take any positive real value ((4.3)), that Γ\X can have any conceivable number of “cusps” ((4.13) and (4.16)), and that these phenomena can occur with π1 (Γ\X) either finitely or infinitely generated. If we drop regularity, then every locally finite connected graph can occur as some Γ\X ((4.17)).


Parabolic Subgroup Tree Lattice Regular Tree Finite Subgroup Finite Graph 
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Copyright information

© Birkhäuser Boston 2001

Authors and Affiliations

  • Hyman Bass
    • 1
  • Alexander Lubotzky
    • 2
  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA
  2. 2.Department of MathematicsHebrew UniversityJerusalemIsrael

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