Counting homomorphisms onto finite solvable groups

Abstract

We present a method for computing the number of epimorphisms from a finitely-presented group G to a finite solvable group \Gamma, which generalizes a formula of G\"aschutz. Key to this approach are the degree 1 and 2 cohomology groups of G, with certain twisted coefficients. As an application, we count low-index subgroups of G. We also investigate the finite solvable quotients of the Baumslag-Solitar groups, the Baumslag parafree groups, and the Artin braid groups.Comment: 30 pages; accepted for publication in the Journal of Algebr

New Permutation Representations of the Braid Group

“…Most results known to date on permutation representations of braids were obtained by Artin ([Ar47]) and by Lin ( [Lin04b]). Some results were also obtained by Matei and Suciu in [MaSu05] where permutation representations of B 3 and B 4 are discussed and computed (we shall address a conjecture made in that paper later on). Let us also mention the work of Arouch in [Ab07] who has applied an algorithmic point of view.…”

“…In section 5 we prove that all type 2 good transitive non-cyclic homomorphisms B k → S 3k are included in the family of homomorphisms we gave in section 3. Finally, in section 6 we give an updated conjecture (updating the conjecture made in [MaSu05]) regarding permutation representations of braid groups and we explain a way to construct homomorphisms B k → S n for n = mk. We give an infinite family of non-cyclic transitive homomorphisms B k → S k(k−1) 2 thus refuting a conjecture made in [MaSu05].…”

Applying TQFT to count regular coverings of Seifert 3-manifolds

“…= #C(x) · δ i,j , the identity #supp(χ i ) · #C(x) = #Γ and (9). Now Z(M N (g; (a 1 , b 1 ), · · · , (a n , b n ))) is equal to the composite…”

“…We choose Γ to be A 5 in the example, because it is a non-solvable finite group, whence beyond the scope of [9].…”

Two paradigms for topological quantum computation

“…So at least for knots, H L (G) is polynomial time computable for G nilpotent. (3) In [31] an algorithm for computing the number of homomorphisms from a given finitely presented group Γ to a finite solvable group is given. It is not clear if this algorithm finishes in polynomial time (in, say, the number of generators of Γ), but it certainly supports the case for (b).…”