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.

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).

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.

HYPERSURFACES IN NONCOMPACT COMPLEX GRASSMANNIANS OF RANK TWO

2012 ◽  
Vol 23 (10) ◽  
pp. 1250103 ◽  
Author(s):  
JÜRGEN BERNDT ◽  
YOUNG JIN SUH
Keyword(s):  
Geometric Structure ◽  
Symmetric Spaces ◽  
Second Fundamental Form ◽  
Kähler Structure ◽  
Noncompact Type ◽  
The Second Fundamental Form ◽  
Special Cases ◽  
Rank One ◽  
Higher Rank ◽  
Riemannian Symmetric Spaces

Consider a Riemannian manifold N equipped with an additional geometric structure, such as a Kähler structure or a quaternionic Kähler structure, and a hypersurface M in N. The geometric structure induces a decomposition of the tangent bundle TM of M into subbundles. A natural problem is to classify all hypersurfaces in N for which the second fundamental form of M preserves these subbundles. This problem is reasonably well understood for Riemannian symmetric spaces of rank one, but not for higher rank symmetric spaces. A general treatment of this problem for higher rank symmetric spaces is out of reach at present, and therefore it is desirable to understand this problem better in a few special cases. Due to some conceptual differences between symmetric spaces of compact type and of noncompact type it appears that one needs to consider these two cases separately. In this paper we investigate this problem for the rank two symmetric space SU 2, m/S(U2Um) of noncompact type.

scholarly journals Novikov Conjectures and Relative Hyperbolicity

1999 ◽  
Vol 85 (2) ◽  
pp. 169 ◽  
Author(s):  
Boris Goldfarb
Keyword(s):  
Symmetric Spaces ◽  
Large Family ◽  
Hyperbolic Groups ◽  
Fundamental Groups ◽  
Arithmetic Groups ◽  
Relatively Hyperbolic Groups ◽  
Rank One ◽  
Higher Rank ◽  
Relatively Hyperbolic ◽  
Relative Hyperbolicity

We consider a class of relatively hyperbolic groups in the sense of Gromov and use an argument modeled after Carlsson-Pedersen to prove Novikov conjectures for these groups. This proof is related to [16,17] which dealt with arithmetic lattices in rank one symmetric spaces and some other arithmetic groups of higher rank. Here whe view the rank one lattices in this different larger context of relativve hyperbolicity which also inclues fundamental groups of pinched hyperbolic manifolds. Another large family of groups from this class is produced using combinatorial hyperbolization techniques.

scholarly journals An Area Theorem for Joint Harmonic Functions on the Product of Homogeneous Trees

2021 ◽  
Author(s):  
Laura Atanasi ◽  
Massimo A. Picardello
Keyword(s):  
Harmonic Functions ◽  
Symmetric Spaces ◽  
Cartesian Product ◽  
Measurable Subset ◽  
Hyperbolic Spaces ◽  
Area Theorem ◽  
Area Function ◽  
Homogeneous Trees ◽  
Rank One ◽  
Higher Rank

AbstractFor harmonic functions v on the disc, it has been known for a long time that non-tangential boundedness a.e.is equivalent to finiteness a.e. of the integral of the area function of v (Lusin area theorem). This result also hold for functions that are non-tangentially bounded only in a measurable subset of the boundary, and has been extended to rank-one hyperbolic spaces, and also to infinite trees (homogeneous or not). No equivalent of the Lusin area theorem is known on higher rank symmetric spaces, with the exception of the degenerate higher rank case given by the cartesian product of rank-one hyperbolic spaces. Indeed, for products of two discs, an area theorem for jointly harmonic functions was proved by M.P. and P. Malliavin, who introduced a new area function; non-tangential boundedness a.e. is a sufficient condition, but not necessary, for the finiteness of this area integral. Their result was later extended to general products of rank-one hyperbolic spaces by Korányi and Putz. Here we prove an area theorem for jointly harmonic functions on the product of a finite number of infinite homogeneous trees; for the sake of simplicity, we give the proofs for the product of two trees. This could be the first step to an area theorem for Bruhat–Tits affine buildings, thereby shedding light on the higher rank continuous set-up.

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