We consider the action of mapping class groups, by preand postcomposition, on branched coverings, and encode them algebraically as mapping class bisets. We show how the mapping class biset of maps preserving a multicurve decomposes into mapping class bisets of smaller complexity, called small mapping class bisets. We phrase the decision problem of Thurston equivalence between branched self-cove...

We give a new characterization of Lusztig’s canonical quotient, a finite group attached to each special nilpotent orbit of a complex semisimple Lie algebra. This group plays an important role in the classification of unipotent representations of finite groups of Lie type. We also define a duality map. To each pair of a nilpotent orbit and a conjugacy class in its fundamental group, the map assi...

Let M be an irreducible projective variety over an algebraically closed field k of characteristic zero equipped with an action of a group Γ. Let EG be a principal G–bundle over M , where G is a connected reductive algebraic group over k, equipped with a lift of the action of Γ on M . We give conditions for EG to admit a Γ–equivariant reduction of structure group to H , where H ⊂ G is a Levi sub...

For a connected complex semi-simple Lie group G and a fixed pair (B, B−) of opposite Borel subgroups of G, we determine when the intersection of a conjugacy class C in G and a double coset BwB− is non-empty, where w is in the Weyl group W of G. The question comes from Poisson geometry, and our answer is in terms of the Bruhat order on W and an involution mC ∈ W associated to C. We study propert...

We give refined estimates for the discrete time and continuous time versions of some basic random walks on the symmetric and alternating groups Sn and An. We consider the following models: random transposition, transpose top with random, random insertion, and walks generated by the uniform measure on a conjugacy class. In the case of random walks on Sn and An generated by the uniform measure on...

We define new subclasses of the class of irreducible sofic shifts. These classes form an infinite hierarchy where the lowest class is the class of almost finite type shifts introduced by B. Marcus. We give effective characterizations of these classes with the syntactic semigroups of the shifts. We prove that these classes define invariants shift equivalence (and thus for conjugacy). Finally, we...

In this paper we prove a combination theorem for Veech subgroups of the mapping class group analogous to the first Klein-Maskit combination theorem for Kleinian groups in which two Fuchsian subgroups are amalgamated along a parabolic subgroup. As a corollary, we construct subgroups of the mapping class group (for all genus at least 2), which are isomorphic to non-abelian closed surface groups i...

The generalized Pauli group and its normalizer, the Clifford group, have a rich mathematical structure which is relevant to the problem of constructing symmetric informationally complete POVMs (SIC-POVMs). To date, almost every known SIC-POVM fiducial vector is an eigenstate of a “canonical” unitary in the Clifford group. I show that every canonical unitary in prime dimensions p > 3 lies in the...

LetN be a normal subgroup of a groupG and let p be a prime. We prove that if the p-part of jx j is a constant for every prime-power order element x 2 N n Z.N /, then N is solvable and has normal p-complement.

Four infinite families of 2-groups are presented, all of whose members possess an outer automorphism that preserves conjugacy classes. The groups in these families are central extensions of their predecessors by a cyclic group of order 2. In particular, for each integer r > 1, there is precisely one 2-group of nilpotency class r in each of the four families. All other known families of 2-groups...