arXiv subjects

Paula Heim

Publications and source records attributed to Paula Heim.

3 recordsLinked to original sources

arXiv2026

Quasimorphisms and Poincar\'e duality in dimension 3

We study $\mathrm{PD}^3$ groups which admit an unbounded quasimorphism to $\mathbb{R}$ with coarsely connected quasikernel. We show that such a group $G$ must either arise as the fundamental group of a torus or Klein-bottle bundle over $S^1$, or be quasiisometric to a Riemannian manifold $(\mathbb{R}^3,g)$ of bounded geometry, with the quasikernel being coarsely equivalent to $\mathbb{H}^2$. If $G$ is moreover hyperbolic, it admits a faithful action on $S^1$ by quasisymmetric homeomorphisms. Our approach features a coarse generalisation of Shapiro's lemma, and a new definition of homological isoperimetric inequalities for metric spaces; these tools make use of Margolis's framework for coarse homological algebra.

math.GR↗

arXiv2026

Realising all countable groups as quasi-isometry groups

Given any countable group $G$, we construct uncountably many quasi-isometry classes of proper geodesic metric spaces with quasi-isometry group isomorphic to $G$. Moreover, if the group $G$ is a hyperbolic group, the spaces we construct are hyperbolic metric spaces. We make use of a rigidity phenomenon for quasi-isometries exhibited by many symmetric spaces, called strong quasi-isometric rigidity. Our method involves the construction of new examples of strongly quasi-isometrically rigid spaces, arising as graphs of strongly quasi-isometrically rigid rank-one symmetric spaces.

math.GR↗

arXiv2024

Tunable two-species spin models with Rydberg atoms in circular and elliptical states

We propose a scheme for constructing versatile quantum simulators using ultracold Rydberg atoms in long-lived circular and elliptical states. By exciting different subspaces of internal atomic states, the atoms can be used to simulate two effective spin species with different spin-spin interactions. The strengths of transverse and longitudinal spin-spin interactions, both intra- and inter-species, can be controlled within a wide range of values. This setup can be used to simulate two-species spin models or lattice models with two sublattices. We show examples of specific models which can be realized.

quant-ph↗