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It has been shown that non-stabilizer eigenstates of permutation gates are appropriate for allowing $d$-dimensional universal quantum computing (uqc) based on minimal informationally complete POVMs. The relevant quantum gates may be built from subgroups of finite index of the modular group $Gamma=PSL(2,mathbb{Z})$ [M. Planat, Entropy 20, 16 (2018)] or more generally from subgroups of fundamental groups of $3$-manifolds [M. Planat, R. Aschheim, M.~M. Amaral and K. Irwin, arXiv 1802.04196(quant-ph)]. In this paper, previous work is encompassed by the use of torsion-free subgroups of Bianchi groups for deriving the quantum gate generators of uqc. A special role is played by a chain of Bianchi congruence $n$-cusped links starting with Thurstons link.
We classify the connected orientable 2-manifolds whose mapping class groups have a dense conjugacy class. We also show that the mapping class group of a connected orientable 2-manifold has a comeager conjugacy class if and only if the mapping class group is trivial.
In this paper we consider convex co-compact subgroups of the projective linear group. We prove that such a group is relatively hyperbolic with respect to a collection of virtually Abelian subgroups of rank two if and only if each open face in the ide
We prove a homological stability theorem for the subgroup of the mapping class group acting as the identity on some fixed portion of the first homology group of the surface. We also prove a similar theorem for the subgroup of the mapping class group
We show that the identity component of the group of diffeomorphisms of a closed oriented surface of positive genus admits many unbounded quasi-morphisms. As a corollary, we also deduce that this group is not uniformly perfect and its fragmentation no
Given a 2-manifold, a fundamental question to ask is which groups can be realized as the isometry group of a Riemannan metric of constant curvature on the manifold. In this paper, we give a nearly complete classification of such groups for infinite-g