We show that every locally compact Polish group is isomorphic to the isometry group of a proper separable metric space. This answers a question of Gao and Kechris. We also analyze the natural action of the isometry group of a separable ultrametric space on the space. This leads us to a structure theorem representing an arbitrary separable ultrametric space as a bundle with an ultrametric base a...

We establish the restricted isometry property for finite dimensional Gabor systems, that is, for families of time–frequency shifts of a randomly chosen window function. We show that the s-th order restricted isometry constant of the associated n×n Gabor synthesis matrix is small provided s ≤ c n/ log n. This improves on previous estimates that exhibit quadratic scaling of n in s. Our proof deve...

The isometry group of a compact n-dimensional hyperbolic man-ifold is known to be finite. We show that for every n ≥ 2, every finite group is realized as the full isometry group of some compact hyperbolic n-manifold. The cases n = 2 and n = 3 have been proven by Greenberg [G] and Ko-jima [K], respectively. Our proof is non constructive: it uses counting results from subgroup growth theory to sh...

Let R be an associative ring with 1 and R the multiplicative group of invertible elements of R. In the paper, subgroups of R which may be regarded as analogues of the similitude group of a non-degenerate sesquilinear reflexive form and of the isometry group of such a form are defined in an abstract way. The main result states that a unipotent abstractly defined similitude must belong to the cor...

We study the large scale geometry of the upper triangular subgroup of PSL2(Z[ 1 n ]), which arises naturally in a geometric context. We prove a quasi-isometry classification theorem and show that these groups are quasi-isometrically rigid with infinite dimensional quasi-isometry group. We generalize our results to a larger class of groups which are metabelian and are higher dimensional analogue...

We consider sub-Riemannian spaces admitting an isometry group that is maximal in the sense any linear of horizontal tangent realized by a global isometry. will show these have canonical partial connection defined on their bundle. However, unlike Riemannian case, such are not uniquely determined curvature and metric cone. Furthermore, number invariants needed to determine model with same cone ge...

We consider the non-square matrix sensing problem, under restricted isometry property (RIP) assumptions. We focus on the non-convex formulation, where any rank-r matrix X ∈ R is represented as UV , where U ∈ R and V ∈ R. In this paper, we complement recent findings on the nonconvex geometry of the analogous PSD setting [5], and show that matrix factorization does not introduce any spurious loca...

Coxeter classified all discrete isometry groups generated by reflections that act on a Euclidean space or on a sphere of an arbitrary dimension (see [1]). His fundamental work became classical long ago. Lobachevsky spaces (classical hyperbolic spaces) are as symmetric as Euclidean spaces and spheres. However, discrete isometry groups generated by reflections, with fundamental polytopes of finit...

For some finite set A of points in R n and some integer k ∈ N we consider the problem of reconstructing the set A up to isometry from the multiset of the |A| k subsets of A of cardinality k given up to isometry. We prove the best possible result for n = 1 and settle an open problem for n = 2 mentioned by Krasikov and Roditty in [8].

The near-horizon geometries of the extremal Kerr black hole and certain generalizations thereof are considered. Their isometry groups are all given by SL(2,R) × U(1). The usual boundary conditions of the Kerr/CFT correspondence enhance the U(1) isometry to a Virasoro algebra. Various alternatives to these boundary conditions are explored. Partial classifications are provided of the boundary con...