The first author has associated in a natural way a profinite group to each irreducible subshift. The group in question was initially obtained as a maximal subgroup of a free profinite semigroup. In the case of minimal subshifts, the same group is shown in the present paper to also arise from geometric considerations involving the Rauzy graphs of the subshift. Indeed, the group is shown to be isomorphic to the inverse limit of the profinite completions of the fundamental groups of the Rauzy graphs of the subshift. A further result involving geometric arguments on Rauzy graphs is a criterion for freeness of the profinite group of a minimal subshift based on the Return Theorem of Berthé et al.

No abstract uploaded!

We define quasi-local conserved quantities in general relativity by using the optimal isometric embedding in Wang and Yau (Commun Math Phys 288(3):919–942, 2009) to transplant Killing fields in the Minkowski spacetime back to the 2-surface of interest in a physical spacetime. To each optimal isometric embedding, a dual element of the Lie algebra of the Lorentz group is assigned. Quasi-local angular momentum and quasi-local center of mass correspond to pairing this element with rotation Killing fields and boost Killing fields, respectively. They obey classical transformation laws under the action of the Poincaré group. We further justify these definitions by considering their limits as the total angular momentum and the total center of mass of an isolated system. These expressions were derived from the Hamilton–Jacobi analysis of the gravitational action and thus satisfy conservation laws. As a result, we obtained an invariant total angular momentum theorem in the Kerr spacetime. For a vacuum asymptotically flat initial data set of order 1, it is shown that the limits are always finite without any extra assumptions. We also study these total conserved quantities on a family of asymptotically flat initial data sets evolving by the vacuum Einstein evolution equation. It is shown that the total angular momentum is conserved under the evolution. For the total center of mass, the classical dynamical formula relating the center of mass, energy, and linear momentum is recovered, in the nonlinear context of initial data sets evolving by the vacuum Einstein evolution equation. The definition of quasi-local angular momentum provides an answer to the second problem in classical general relativity on Penrose’s list (Proc R Soc Lond Ser A 381(1780):53–63, 1982).

No abstract uploaded!

Let $\U_q(\fg)$ be the quantum supergroup of $\gl_{m|n}$ or the modified quantum supergroup of $\osp_{m|2n}$ over the field of rational functions in $q$, and let $V_q$ be the natural module for $\U_q(\fg)$. There exists a unique tensor functor, associated with $V_q$, from the category of ribbon graphs to the category of finite dimensional representations of $\U_q(\fg)$, which preserves ribbon category structures. We show that this functor is full in the cases $\fg=\gl_{m|n}$ or $\osp_{2\ell+1|2n}$. For $\fg=\osp_{2\ell|2n}$, we show that the space $\Hom_{\U_q(\fg)}(V_q^{\otimes r}, V_q^{\otimes s})$ is spanned by images of ribbon graphs if $r+s< 2\ell(2n+1)$. The proofs involve an equivalence of module categories for two versions of the quantisation of $\U(\fg)$.