TY - JOUR
T1 - (Non)-escape of mass and equidistribution for horospherical actions on trees
AU - Ciobotaru, Corina
AU - Finkelshtein, Vladimir
AU - Sert, Cagri
N1 - Funding Information:
The authors are thankful to Marc Burger and Manfred Einsiedler for helpful discussions. The authors also thank an anonymous referee for a careful reading, several remarks clarifying the exposition and helpful bibliographical suggestions. V.F. is supported by ERC Consolidator grant 648329 (GRANT). C.S. is supported by SNF grants 178958 and 182089.
Publisher Copyright:
© 2021, The Author(s).
PY - 2022/2
Y1 - 2022/2
N2 - Let G be a large group acting on a biregular tree T and Γ ≤ G a geometrically finite lattice. In an earlier work, the authors classified orbit closures of the action of the horospherical subgroups on G/ Γ. In this article we show that there is no escape of mass and use this to prove that, in fact, dense orbits equidistribute to the Haar measure on G/ Γ. On the other hand, we show that new dynamical phenomena for horospherical actions appear on quotients by non-geometrically finite lattices: we give examples of non-geometrically finite lattices where an escape of mass phenomenon occurs and where the orbital averages along a Følner sequence do not converge. In the last part, as a by-product of our methods, we show that projections to Γ \ T of the uniform distributions on large spheres in the tree T converge to a natural probability measure on Γ \ T. Finally, we apply this equidistribution result to a lattice point counting problem to obtain counting asymptotics with exponential error term.
AB - Let G be a large group acting on a biregular tree T and Γ ≤ G a geometrically finite lattice. In an earlier work, the authors classified orbit closures of the action of the horospherical subgroups on G/ Γ. In this article we show that there is no escape of mass and use this to prove that, in fact, dense orbits equidistribute to the Haar measure on G/ Γ. On the other hand, we show that new dynamical phenomena for horospherical actions appear on quotients by non-geometrically finite lattices: we give examples of non-geometrically finite lattices where an escape of mass phenomenon occurs and where the orbital averages along a Følner sequence do not converge. In the last part, as a by-product of our methods, we show that projections to Γ \ T of the uniform distributions on large spheres in the tree T converge to a natural probability measure on Γ \ T. Finally, we apply this equidistribution result to a lattice point counting problem to obtain counting asymptotics with exponential error term.
KW - Automorphisms of trees
KW - Equidistribution
KW - Escape of mass
KW - Non-linear homogeneous dynamics
UR - http://www.scopus.com/inward/record.url?scp=85113821304&partnerID=8YFLogxK
U2 - 10.1007/s00209-021-02852-1
DO - 10.1007/s00209-021-02852-1
M3 - Journal article
AN - SCOPUS:85113821304
SN - 0025-5874
VL - 300
SP - 1673
EP - 1704
JO - Mathematische Zeitschrift
JF - Mathematische Zeitschrift
IS - 2
ER -