We prove that free pre-Lie algebras, when considered as Lie algebras, are free. Working in the category of S-modules, we define a natural filtration on the space of generators. We also relate the symmetric group action on generators with the structure of the anticyclic PreLie operad.

Suppose that G is a linearly reductive group. Good degree bounds for generators of invariant rings were given in [2]. Here we study the minimal free resolution of the invariant ring. Recently it was shown that if G is a finite linearly reductive group, then the ring of invariants is generated in degree ≤ |G| (see [5, 6, 3]). This extends the classical result of Noether who proved the bound in c...

In the study of Fuchsian groups, it is a nontrivial problem to determine a set of generators. Using a dynamical approach we construct for any cocompact arithmetic Fuchsian group a fundamental region in SL2(R) from which we determine a set of small generators.

We give an elementary proof that there are two topological generators for the full group of every aperiodic hyperfinite probability measure preserving Borel equivalence relation. Our proof explicitly constructs topological generators for the orbit equivalence relation of the irrational rotation of the circle, and then appeals to Dye’s theorem and a Baire category argument to conclude the genera...

We find new bounds on the minimal number of generators of crystallographic groups with p-group holonomy. We also show that similar bounds exist on the minimal number of generators of the abelianizations of arbitrary crystallographic groups. As a consequence, we show that this restricts the rank of elementary abelian p-groups that can act effectively on closed connected flat orbifolds.

A presentation of a group with two generators having unsolvable word problem and an explicit countable presentation of Mihailova subgroup of F2 × F2 with finite number of generators are given. Where Mihailova subgroup of F2 × F2 enjoys the unsolvable subgroup membership problem. One then can use the presentation to create entities’ private keys in a public key cryptsystem.

Abstract. The Heisenberg group is one of the simplest sub-Riemannian settings in which we can define non-elliptic Hörmander type generators. We can then consider coercive inequalities associated to such generators. We prove that a certain class of nontrivial Gibbs measures with quadratic interaction potential on an infinite product of Heisenberg groups satisfy logarithmic Sobolev inequalities.

We provide a simple criterion for an element of the mapping class group closed surface to be normal generator group. apply this show that every nontrivial periodic is not hyperelliptic involution when genus at least 3. also give many examples pseudo-Anosov generators, answering question D. Long. In fact we with stretch factor less than $\\sqrt{2}$ generator. Even more, generators arbitrarily la...