scholarly journals Correction: Gušić, D. Prime Geodesic Theorems for Compact Locally Symmetric Spaces of Real Rank One. Mathematics 2020, 8, 1762

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 710
Author(s):  
Dženan Gušić
Keyword(s):  
Symmetric Spaces ◽  
Real Rank ◽  
Locally Symmetric Spaces ◽  
Rank One

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

On the logarithmic derivative of zeta functions for compact even-dimensional locally symmetric spaces of real rank one

2019 ◽  
Vol 69 (2) ◽  
pp. 311-320 ◽  
Author(s):  
Muharem Avdispahić ◽  
Dženan Gušić
Keyword(s):  
Symmetric Spaces ◽  
Logarithmic Derivative ◽  
Zeta Functions ◽  
Real Rank ◽  
Locally Symmetric Spaces ◽  
Rank One ◽  
Approximate Formulas

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.

scholarly journals On Some Classical andWeighted Estimates for SL4

2020 ◽  
Vol 15 ◽  
Keyword(s):  
Error Term ◽  
Symmetric Spaces ◽  
Compact Riemann Surface ◽  
Real Rank ◽  
Classical Approach ◽  
Locally Symmetric Spaces ◽  
Compact Symmetric Space ◽  
Rank One ◽  
Prime Geodesic Theorem ◽  
Correct Estimate

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.

scholarly journals On the Growth of L2-Invariants of Locally Symmetric Spaces, II: Exotic Invariant Random Subgroups in Rank One

2018 ◽  
Vol 2020 (9) ◽  
pp. 2588-2625
Author(s):  
Miklos Abert ◽  
Nicolas Bergeron ◽  
Ian Biringer ◽  
Tsachik Gelander ◽  
Nikolay Nikolov ◽  
...  
Keyword(s):  
Asymptotic Behavior ◽  
Lie Groups ◽  
Symmetric Spaces ◽  
Betti Numbers ◽  
Real Rank ◽  
Spectral Invariants ◽  
Locally Symmetric Spaces ◽  
Rank One ◽  
Higher Rank ◽  
Invariant Random Subgroups

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).

The spectrum and heat dynamics of locally symmetric spaces of higher rank

2014 ◽  
Vol 35 (5) ◽  
pp. 1524-1545 ◽  
Author(s):  
LIZHEN JI ◽  
ANDREAS WEBER
Keyword(s):  
Symmetric Spaces ◽  
Chaotic Behavior ◽  
Heat Semigroup ◽  
Locally Symmetric Spaces ◽  
Rank One ◽  
Higher Rank

The aim of this paper is to study the spectrum of the$L^{p}$Laplacian and the dynamics of the$L^{p}$heat semigroup on non-compact locally symmetric spaces of higher rank. Our work here generalizes previously obtained results in the setting of locally symmetric spaces of rank one to higher rank spaces. Similarly as in the rank-one case, it turns out that the$L^{p}$heat semigroup on$M$has a certain chaotic behavior if$p\in (1,2)$, whereas for$p\geq 2$such chaotic behavior never occurs.

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