Related papers: Relating two genus 0 problems of John Thompson

H. Davenport's Problem asks: What can we expect of two polynomials, over the integers, with the same ranges on almost all residue class fields? This stood out among many separated variable problems posed by Davenport, D.J. Lewis and A.…

Let $G$ be an additive finite abelian group. A sequence over $G$ is called a minimal zero-sum sequence if the sum of its terms is zero and no proper subsequence has this property. Davenport's constant of $G$ is the maximum of the lengths of…

Evidences have suggested that counting representations are sometimes tractable even when the corresponding classification problem is almost impossible, or "wild" in a precise sense. Such counting problems are directly related to matrix…

A remarkable result of Thompson states that a finite group is soluble if and only if its two-generated subgroups are soluble. This result has been generalized in numerous ways, and it is in the core of a wide area of research in the theory…

The Davenport constant for a finite abelian group $G$ is the minimal length $\ell$ such that any sequence of $\ell$ terms from $G$ must contain a nontrivial zero-sum sequence. For the group $G=(\mathbb Z/n\mathbb Z)^2$, its value is $2n-1$,…

We study the monodromy groups of compositions of two indecomposable polynomials. In particular, we show that such monodromy groups either fulfill a certain "largeness" property, or are in an explicit list of exceptions. Such largeness…

We will say that a group G possesses the Magnus property if for any two elements u,v in G with the same normal closure, u is conjugate to v or v^{-1}. We prove that some one-relator groups, including the fundamental groups of closed…

All groups are 2-generator. For any prime-power q, Theorem 1 constructs a solvable matrix group over a quotient of a Laurent polynomial ring. This group is closely related to a group of exponent q as shown in Theorems 2 & 3 . Theorem 4 in…

In a previous paper, the author and his collaborators studied the phenomenon of isotropy in the context of single-sorted equational theories, and showed that the isotropy group of the category of models of any such theory encodes a notion…

For a finite abelian group $G$ and a positive integer $k$, let $\mathsf{D}_k(G)$ denote the smallest integer $\ell$ such that each sequence over $G$ of length at least $\ell$ has $k$ disjoint nontrivial zero-sum subsequences. It is known…

Let $R$ be a regular semi-local ring, essentially of finite type over an infinite perfect field of characteristic $p \ge 3$. We show that the cycle class map with modulus from an earlier work of the authors induces a pro-isomorphism between…

We present a general method for introducing finitely axiomatizable "minimal" two-sorted theories for various subclasses of P (problems solvable in polynomial time). The two sorts are natural numbers and finite sets of natural numbers. The…

We show that one can naturally describe elements of R. Thompson's finitely presented infinite simple group $V$, known by Thompson to have a presentation with four generators and fourteen relations, as products of permutations analogous to…

We describe a procedure for constructing a generalized Thompson group out of a family of groups that is equipped with what we call a cloning system. The previously known Thompson groups F, V, Vbr and Fbr arise from this procedure using,…

We continue with the functional approach to the P-versus-NP problem, begun in [2, 3]. We previously constructed a monoid RM^P that is non-regular iff NP is not P. We now construct homomorphic images of RM^P with interesting properties. In…

We show how to construct a family of groups with simple commutator subgroups from aperiodic 1-vertex, finitely aligned higher rank graphs (which are, in fact, a class of cancellative monoids). Inverse semigroups form the intermediary…

We consider a combinatorial problem occurring naturally in a group theoretical setting and provide a constructive solution in a special case. More precisely, in 1999 the author established a logarithmic bound for the derived length of the…

The counting of partitions according to their genus is revisited. The case of genus 0 -- non-crossing partitions -- is well known. Our approach relies on two pillars: first a functional equation between generating functions, originally…

In this talk we consider the relationship between the conjectured uniqueness of the Moonshine module of Frenkel, Lepowsky and Meurman and Monstrous Moonshine, the genus zero property for Thompson series discovered by Conway and Norton. We…

The groups G_{k,1} of Richard Thompson and Graham Higman can be generalized in a natural way to monoids, that we call M_{k,1}, and to inverse monoids, called Inv_{k,1}; this is done by simply generalizing bijections to partial functions or…