Stability phenomena for Martin boundaries of relatively hyperbolic groups
- PDF / 708,671 Bytes
- 59 Pages / 439.37 x 666.142 pts Page_size
- 69 Downloads / 180 Views
Stability phenomena for Martin boundaries of relatively hyperbolic groups Matthieu Dussaule1
· Ilya Gekhtman2
Received: 24 October 2019 / Revised: 28 June 2020 / Accepted: 8 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Let be a relatively hyperbolic group and let μ be an admissible symmetric finitely supported probability measure on . We extend Floyd–Ancona type inequalities from Gekhtman et al. (Martin boundary covers Floyd boundary, 2017. arXiv:1708.02133) up to the spectral radius R of μ. We use them to find the precise homeomorphism type of the r -Martin boundary, which describes r -harmonic functions, for every r ≤ R. We also define a notion of spectral degeneracy along parabolic subgroups which is crucial to describe the homeomorphism type of the R-Martin boundary. Finally, we give a criterion for (strong) stability of the Martin boundary in the sense of Picardello and Woess (in: Potential theory, de Gruyter, 1992) in terms of spectral degeneracy. We then prove that this criterion is always satisfied in small rank, so that in particular, the Martin boundary of an admissible symmetric finitely supported probability measure on a geometrically finite Kleinian group of dimension at most 5 is always strongly stable. Mathematics Subject Classification Primary 20F67 · 31C35; Secondary 20F65 · 60B15
1 Introduction 1.1 Martin boundaries and stability We consider a finitely generated group together with a probability measure μ. The μ-random walk starting at e on is defined as X n = g1 . . . gn , where the gk are independent, identically distributed according to μ random variables on . We always assume that the random walk is admissible: the support of μ generates the group as
B
Matthieu Dussaule [email protected]
1
Universite de Nantes, Nantes, France
2
Technion, Haifa, Israel
123
M. Dussaule, I. Gekhtman
a semi-group. In many situations, one can understand the asymptotic behaviour of X n in terms of geometric properties at large scale of . One way to do so is to try to compare geometric boundaries of the group, encoding how geodesics behave at infinity, with probabilistic boundaries, encoding how the random walk behaves asymptotically. In the groups we will study here, the random walk will always be transient, meaning that it almost surely goes to infinity. One can then define the Martin boundary as follows. Define the Green function G(·, ·) by G(x, y) =
μ∗n (x −1 y),
n≥0
where μ∗n is the nth convolution power of μ. Precisely, μ∗n =
z 0 =x,z 1 ,...,z n =y
−1 μ(z 0−1 z 1 )μ(z 1−1 z 2 ) . . . μ(z n−1 z n ).
Define then the Martin kernel K (·, ·) by K (x, y) =
G(x, y) . G(e, y)
We endow with the discrete topology. The Martin compactification of and μ is the smallest compact metrizable set X such that embeds as a dense and open set in X and such that the function K (·, ·) extends as a continuous function on × X . In other words, a sequence gn in converges to a point in the Martin compactification if and only if for every
Data Loading...
