We prove that PSL2(Z[ 1 p ]) gives the first example of groups which are not quasi-isometric to each other but have the same quasi-isometry group. Namely, PSL2(Z[ 1 p ]) and PSL2(Z[ 1 q ]) are not quasi-isometric unless p = q, and, independent of p, the quasi-isometry group of PSL2(Z[ 1 p ]) is PSL2(Q). In addition, we characterize PSL2(Z[ 1 p ]) uniquely among all finitely generated groups by ...

Recall that the quasi-isometry group QI(X) of a metric space X is the set of equivalence classes of quasi-isometries f : X → X, where two quasiisometries f1, f2 are equivalent iff supx d(f1(x), f2(x)) <∞ (here we consider G as a metric space with a word metric). One approach to this question, which has been the most successful one, is to find an ‘optimal’ space X quasi-isometric to G and show t...

A classical theorem of Hurewitz says that the isometry group of a closed 2-dimensional hyperbolic manifold acts faithfully on its first homology group. The analogous theorem in dimension 3 is false. In this paper we consider the class of 3-manifolds which are cyclic branched coverings of knots in the 3-sphere S. We characterize the isometry group actions which are homologically faithful in the ...

We show that there is a remarkable connection between the harmonic superspace (HSS) formulation of N = 2, d = 4 supersymmetric quaternionic Kähler sigma models that couple to N = 2 supergravity and the minimal unitary representations of their isometry groups. In particular, for N = 2 sigma models with quaternionic symmetric target spaces of the form G/H × SU(2) we establish a one-to-one mapping...

‎In this paper‎, ‎first we develop the duality concept for $g$-Bessel sequences‎ ‎and Bessel fusion sequences in Hilbert spaces‎. ‎We obtain some results about dual‎, ‎pseudo-dual ‎and approximate dual of frames and fusion frames‎. ‎We also expand every $g$-Bessel ‎sequence to a frame by summing some elements‎. ‎We define the restricted isometry property for ‎$g$-frames and generalize some resu...

Definition 1 (Restricted isometry and orthogonality). The S-restricted isometry constant δS of a matrix F ∈ Rn×m is the smallest quantity such that (1− δS)‖x‖2 ≤ ‖FTx‖2 ≤ (1 + δS)‖x‖2 for all T ⊆ [m] with |T | ≤ S and all x ∈ R|T |. The (S, S′)-restricted orthogonality constant θS,S′ of F is the smallest quantity such that |FTx · FT ′x′| ≤ θS,S′‖c‖‖c‖ for all disjoint T, T ′ ⊆ [m] with |T | ≤ S...

In our previous work [4] we showed that the extension property holds for linear codes over a finite module alphabet with respect to the symmetrized weight composition if and only if the socle of the alphabet is cyclic. For alphabets with a noncyclic socle we proved the existence of an unextendable isometry. However, our construction was implicit. This paper deals with a particular case of an al...

In this paper the author gives necessary and sufficient conditions under which to a stable weak isometry f in a directed group G there exists a direct decomposition G = A × B of G such that f(x) = x(A) − x(B) for each x ∈ G. Further, some results on weak isometries in partially ordered groups are established. Isometries in an abelian lattice ordered group (l-group) have been introduced and inve...

We show how the supersymmetric properties of three dimensional black holes can be obtained algebraically. The black hole solutions are constructed as quotients of the supergroup OSp(1| 2;R) by a discrete subgroup of its isometry supergroup. The generators of the action of the isometry supergroup which commute with these identifications are found. These yield the supersymmetries for the black ho...

We consider the behavior of piecewise isometries in Euclidean spaces. We show that if n is odd and the system contains no orientation reversing isometries then recurrent orbits with rational coding are not expected. More precisely, a prevalent set of piecewise isometries do not have recurrent points having rational coding. This implies that when all atoms are convex no periodic points exist for...