We show that the universal central extensions of the little projective group of any Moufang polygon is precisely the Steinberg group obtained from its defining commutator relations, provided the defining structure is not too small. As an application, we get that also the universal central extensions of the little projective group of any 2-spherical Moufang twin building is precisely the Steinbe...

Finite group extensions offer a natural language to quantum computing. In a nutshell, one roughly describes the action of a quantum computer as consisting of two finite groups of gates: error gates from the general Pauli group P and stabilizing gates within an extension group C. In this paper one explores the nice adequacy between group theoretical concepts such as commutators, normal subgroups...

In 1957 D.R. Hughes published the following problem in group theory. Let G be a group and p a prime. Define Hp(G) to be the subgroup of G generated by all the elements of G which do not have order p . Is the following conjecture true: either Hp(G) = 1, Hp(G) = G , or [G : Hp(G)] = p? This conjecture has become known as the Hughes conjecture. After various classes of groups were shown to satisfy...

A conjecture of Gromov states that a one-ended word-hyperbolic group must contain a subgroup that is isomorphic to the fundamental group of a closed hyperbolic surface. Recent papers by Gordon and Wilton and by Kim and Wilton give sufficient conditions for hyperbolic surface groups to be embedded in a hyperbolic Baumslag double G. Using Nielsen cancellation methods based on techniques from prev...

• Coxeter groups with respect to the standard generating set [5, p. 45, Exercise 26], the argument being that of Solomon [18]; • virtually abelian groups with respect to any generating set [4]; • two-step nilpotent groups with infinite cyclic commutator subgroup with respect to some generating set [19]. (On the other hand, it is shown in [19] that if G has Heisenberg rank at least 2, then G has...

We show that among Cr extensions (r > 0) of a uniformly hyperbolic dynamical system with fiber the standard real Heisenberg group Hn of dimension 2n + 1, those that avoid an obvious obstruction to topological transitivity are generically topological transitive. Moreover, if one considers extensions with fiber a connected nilpotent Lie group with compact commutator subgroup (for example, Hn/Z), ...

We study two group theoretic problems, Group Intersection and Double Coset Membership, in the setting of black-box groups, where Double Coset Membership generalizes a set of problems, including Group Membership, Group Factorization, and Coset Intersection. No polynomial-time classical algorithms are known for these problems. We show that for solvable groups, there exist efficient quantum algori...

Anshel, et. al., introduced a new cryptographic protocol, the Commutator key agreement protocol, whose strength lies heavily on the difficulty of the generalized conjugacy problem in subgroups of the Braid group. A natural approach to this problem is by using a length-based method, with the length of the Garside normal form as a length function. Experiments show that this approach is far from b...

We prove that the stable commutator lengths of Dehn twists are positive, implying that the natural map from the second bounded cohomology to the ordinary cohomology is not injective, that mapping class groups are not uniformly perfect and that the growth rate of Dehn twists are linear. We then prove that the dimensions of the second bounded cohomology of hyperelliptic mapping class groups and o...