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69 results

Page 1 of 4

  • Cusps, Kleinian groups, and Eisenstein series

  • Part of
  • Beibei Liu, Shi Wang
  • Journal:
    Forum of Mathematics, Sigma / Volume 11 / 2023
    Published online by Cambridge University Press:
    31 August 2023, e75
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  • We study the Eisenstein series associated to the full rank cusps in a complete hyperbolic manifold. We show that given a Kleinian group $\Gamma <{\operatorname{\mathrm{Isom}}}^+(\mathbb H^{n+1})$, each full rank cusp corresponds to a cohomology class in $H^{n}(\Gamma , V)$, where V is either the trivial coefficient or the adjoint representation. Moreover, by computing the intertwining operator, we show that different cusps give rise to linearly independent classes.

  • Uniform spectral gap and orthogeodesic counting for strong convergence of Kleinian groups

  • Part of
  • Beibei Liu, Franco Vargas Pallete
  • Journal:
    Forum of Mathematics, Sigma / Volume 11 / 2023
    Published online by Cambridge University Press:
    04 August 2023, e68
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  • We show convergence of small eigenvalues for geometrically finite hyperbolic n-manifolds under strong limits. For a class of convergent convex sets in a strongly convergent sequence of Kleinian groups, we use the spectral gap of the limit manifold and the exponentially mixing property of the geodesic flow along the strongly convergent sequence to find asymptotically uniform counting formulas for the number of orthogeodesics between the convex sets. In particular, this provides asymptotically uniform counting formulas (with respect to length) for orthogeodesics between converging Margulis tubes, geodesic loops based at converging basepoints, and primitive closed geodesics.

Page 1 of 4