34,815 research outputs found
Covering the alternating groups by products of cycle classes
AbstractGiven integers k,l⩾2, where either l is odd or k is even, we denote by n=n(k,l) the largest integer such that each element of An is a product of k cycles of length l. For an odd l, k is the diameter of the undirected Cayley graph Cay(An,Cl), where Cl is the set of all l-cycles in An. We prove that if k⩾2 and l⩾9 is odd and divisible by 3, then 23kl⩽n(k,l)⩽23kl+1. This extends earlier results by Bertram [E. Bertram, Even permutations as a product of two conjugate cycles, J. Combin. Theory 12 (1972) 368–380] and Bertram and Herzog [E. Bertram, M. Herzog, Powers of cycle-classes in symmetric groups, J. Combin. Theory Ser. A 94 (2001) 87–99]
Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks
AbstractFuchsian groups (acting as isometries of the hyperbolic plane) occur naturally in geometry, combinatorial group theory, and other contexts. We use character-theoretic and probabilistic methods to study the spaces of homomorphisms from Fuchsian groups to symmetric groups. We obtain a wide variety of applications, ranging from counting branched coverings of Riemann surfaces, to subgroup growth and random finite quotients of Fuchsian groups, as well as random walks on symmetric groups. In particular, we show that, in some sense, almost all homomorphisms from a Fuchsian group to alternating groups An are surjective, and this implies Higman's conjecture that every Fuchsian group surjects onto all large enough alternating groups. As a very special case, we obtain a random Hurwitz generation of An, namely random generation by two elements of orders 2 and 3 whose product has order 7. We also establish the analogue of Higman's conjecture for symmetric groups. We apply these results to branched coverings of Riemann surfaces, showing that under some assumptions on the ramification types, their monodromy group is almost always Sn or An. Another application concerns subgroup growth. We show that a Fuchsian group Γ has (n!)μ+o(1) index n subgroups, where μ is the measure of Γ, and derive similar estimates for so-called Eisenstein numbers of coverings of Riemann surfaces. A final application concerns random walks on alternating and symmetric groups. We give necessary and sufficient conditions for a collection of ‘almost homogeneous’ conjugacy classes in An to have product equal to An almost uniformly pointwise. Our methods involve some new asymptotic results for degrees and values of irreducible characters of symmetric groups
Simple endotrivial modules for quasi-simple groups
We investigate simple endotrivial modules of finite quasi-simple groups and
classify them in several important cases. This is motivated by a recent result
of Robinson showing that simple endotrivial modules of most groups come from
quasi-simple groups.Comment: 31 pages. Changes from (v1): in Theorem 1.3, we removed the
assumption that G<SL and proved that endotrivial modules are liftable to
characteristic zero in all generality. (v3): revised version, to appear in
Journal f\"ur die Reine und Angewandte Mathemati
Critical classes, Kronecker products of spin characters, and the Saxl conjecture
Highlighting the use of critical classes, we consider constituents in
Kronecker products, in particular of spin characters of the double covers of
the symmetric and alternating groups. We apply results from the spin case to
find constituents in Kronecker products of characters of the symmetric groups.
Via this tool, we make progress on the Saxl conjecture; this claims that for a
triangular number , the square of the irreducible character of the symmetric
group labelled by the staircase contains all irreducible characters of
as constituents. We find a large number of constituents in this square
which were not detected by other methods. Moreover, the investigation of
Kronecker products of spin characters inspires a spin variant of Saxl's
conjecture.Comment: 17 page
Surface bundles over surfaces: new inequalities between signature, simplicial volume and Euler characteristic
We present three new inequalities tying the signature, the simplicial volume
and the Euler characteristic of surface bundles over surfaces. Two of them are
true for any surface bundle, while the third holds on a specific family of
surface bundles, namely the ones that arise through a ramified covering. These
are the main known examples of bundles with non-zero signature.Comment: 14 pages. Simplified the proof of Proposition 1.2. This is the final
version, accepted in Geometriae Dedicat
- …