scholarly journals On the Poisson Transform on Symmetric Spaces of Real Rank One

2000 ◽  
Vol 174 (2) ◽  
pp. 513-523 ◽  
Author(s):  
Alexandru D. Ionescu
2019 ◽  
Vol 69 (2) ◽  
pp. 311-320 ◽  
Author(s):  
Muharem Avdispahić ◽  
Dženan Gušić

Abstract We derive approximate formulas for the logarithmic derivative of the Selberg and the Ruelle zeta functions over compact, even-dimensional, locally symmetric spaces of real rank one. The obtained formulas are given in terms of zeta singularities.


The purpose of this paper is two-sided. First, we obtain the correct estimate of the error term in the classical prime geodesic theorem for compact symmetric space SL4. As it turns out, the corrected error term depends on the degree of a certain polynomial appearing in the functional equation of the attached zeta function. This is in line with the known result in the case of compact Riemann surface, or more generally, with the corresponding result in the case of compact locally symmetric spaces of real rank one. Second, we derive a weighted form of the theorem. In particular, we prove that the aforementioned error term can be significantly improved when the classical approach is replaced by its higher level analogue.


Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 710
Author(s):  
Dženan Gušić

The author wishes to make the following correction to the paper [...]


2018 ◽  
Vol 2020 (9) ◽  
pp. 2588-2625
Author(s):  
Miklos Abert ◽  
Nicolas Bergeron ◽  
Ian Biringer ◽  
Tsachik Gelander ◽  
Nikolay Nikolov ◽  
...  

Abstract In the 1st paper of this series we studied the asymptotic behavior of Betti numbers, twisted torsion, and other spectral invariants for sequences of lattices in Lie groups G. A key element of our work was the study of invariant random subgroups (IRSs) of G. Any sequence of lattices has a subsequence converging to an IRS, and when G has higher rank, the Nevo–Stuck–Zimmer theorem classifies all IRSs of G. Using the classification, one can deduce asymptotic statements about spectral invariants of lattices. When G has real rank one, the space of IRSs is more complicated. We construct here several uncountable families of IRSs in the groups SO(n, 1), n ≥ 2. We give dimension-specific constructions when n = 2, 3, and also describe a general gluing construction that works for every n. Part of the latter construction is inspired by Gromov and Piatetski-Shapiro’s construction of non-arithmetic lattices in SO(n, 1).


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