Simplical volume of closed locally symmetric spaces of non-compact type

We show that closed, locally symmetric spaces of non-compact type have positive simplicial volume. This gives a positive answer to a question that was first raised by (Gromov in Inst Hautes études Sci Publ Math 56:5, 1982).     

On mappings of \mathbb{Q}^d to \mathbb{Q}^d that preserve distances 1 and \sqrt 2 and the Beckman-Quarles Theorem

Benz proved that every mapping that preserves the distances 1 and 2 is an isometry, provided d ≥ 5. We prove that every mapping that preserves the distances 1 and is an isometry, provided d ≥ 5.     

Scattered subsets of\mathbb{Q}

A linear order isscattered if it contains no copy of the rationals. The scattered subsets of form a better quasi-order under embeddability via order automorphisms.Research supported by NSERC grant #9848.     

Finite energy and totally geodesic maps from locally symmetric spaces of finite volume

LetM=G/K be a locally symmetric space of finite volume and rank 2. We show that any map fromM of weighted finite energy in the sense of Saper can be deformed into a finite energy map. As a consequence such maps can be deformed into totally geodesic ones, and a geometric generalization of Margulis' superrigidity theorem is obtained.     

Hyperelliptic parametrizations of $$\pmb {\mathbb {Q}}$$ Q -curves

The Ramanujan Journal - In this paper, our aim is to find the radii of starlikeness and convexity of the Ramanujan type function for three different kinds of normalization by using their...     

Virtual Betti numbers of compact locally symmetric spaces

We show that the virtual Betti number of a compact locally symmetric space with arithmetic fundamental group is either the Betti number of the compact dual or else is infinite.     

A vanishing theorem for modular symbols on locally symmetric spaces

A modular symbol is the fundamental class of a totally geodesic submanifold embedded in a locally Riemannian symmetric space , which is defined by a subsymmetric space . In this paper, we consider the modular symbol defined by a semisimple symmetric pair (G,G'), and prove a vanishing theorem with respect to the -component in the Matsushima-Murakami formula based on the discretely decomposable theorem of the restriction . In particular, we determine explicitly the middle Hodge components of certain totally real modular symbols on the locally Hermitian symmetric spaces of type IV. Received: December 8, 1996     

Pointwise bounds for L 2 eigenfunctions on locally symmetric spaces

We prove pointwise bounds for L ² eigenfunctions of the Laplace-Beltrami operator on locally symmetric spaces with -rank one if the corresponding eigenvalues lie below the continuous part of the L ² spectrum. Furthermore, we use these bounds in order to obtain some results concerning the L ^p spectrum.     

A Haar component for quantum limits on locally symmetric spaces

We prove lower bounds for the entropy of limit measures associated to non-degenerate sequences of eigenfunctions on locally symmetric spaces of non-positive curvature. In the case of certain compact quotients of the space of positive definite n × n matrices (any quotient for n = 3, quotients associated to inner forms in general), measure classification results then show that the limit measures must have a Haar component. This is consistent with the conjecture that the limit measures are absolutely continuous.     

On the growth of Betti numbers of locally symmetric spaces

We announce new results concerning the asymptotic behavior of the Betti numbers of higher rank locally symmetric spaces as their volumes tend to infinity. Our main theorem is a uniform version of the Lück Approximation Theorem (Lück, 1994 [10]) which is much stronger than the linear upper bounds on Betti numbers given by Gromov in Ballmann et al. (1985) [3].The basic idea is to adapt the theory of local convergence, originally introduced for sequences of graphs of bounded degree by Benjamini and Schramm, to sequences of Riemannian manifolds. Using rigidity theory we are able to show that when the volume tends to infinity, the manifolds locally converge to the universal cover in a sufficiently strong manner that allows us to derive the convergence of the normalized Betti numbers.     

Simplicial volume of locally symmetric spaces covered by $${\rm SL}_{3}{\mathbb{R}}/{\rm SO}(3)$$

We give the first complete proof of the strict positivity of the simplicial volume of compact locally symmetric spaces covered by and show why the proof in [Savage RP, Trans Amer Math Soc 274(1):239–263, 1982] is incorrect.      

Moduli spaces of holomorphic mappings into hyperbolically imbedded complex spaces and locally symmetric spaces

Dedicated to professor K. Oikawa on his 60th birthday     

Classification of invariant AHS-structures on semisimple locally symmetric spaces

We discuss which semisimple locally symmetric spaces admit an AHS-structure invariant under local symmetries. We classify them for all types of AHS-structures and determine possible equivalence classes of such AHS-structures.