We explore algebraic subgroups of the Cremona group Cn over an algebraically closed field of characteristic zero. First, we consider some class of algebraic subgroups of Cn that we call flattenable. It contains all tori. Linearizability of the natural rational actions of flattenable subgroups on An is intimately related to rationality of the invariant fields and, for tori, is equivalent to it. ...

Using classification of finite simple groups, I show that a finite subgroup G of GL(n)(C), where C = the complex numbers, contains a commutative normal subgroup M of index at most (n + 1)!n(alogn+b). Moreover, if G is primitive and does not contain normal subgroups that are direct products of large alternating groups, then the factor (n + 1)! can be dropped. I further show that similar statemen...

Abstract. We establish the lower bound p2t(e, e) exp(−t1/3), for the large times asymptotic behaviours of the probabilities p2t(e, e) of return to the origin at even times 2t, for random walks associated with finite symmetric generating sets of solvable groups of finite Prüfer rank. (A group has finite Prüfer rank if there is an integer r , such that any of its finitely generated subgroup admit...

An operator $I$ on a real Lie algebra $A$ is called complex structure if $I^2=-Id$ and the $\sqrt{-1}$-eigenspace $A^{1,0}$ subalgebra in complexification of $A$. A hypercomplex triple structures $I,J$ $K$ satisfying quaternionic relations. We call nilpotent quaternionic-solvable there exists finite filtration by quaternionic-invariant subalgebras with commutative subquotients which converges t...

A group is small if it has countably many complete n-types over the empty set for each natural number n. More generally, a group G is weakly small if it has countably many complete 1-types over every finite subset of G. We show here that in a weakly small group, subgroups which are definable with parameters lying in a finitely generated algebraic closure satisfy the descending chain conditions ...

We obtain a strong law of large numbers and a functional central limit theorem, as t → ∞, for the number of records up to time t and the Lebesgue measure (length) of the subset of the time interval [0, t] during which the Poisson process is in a record lifetime. strong law of large numbers.

Let G be a nite group and let i (G) denote the proportion of ordered pairs of G that generate a subgroup of nilpotency class i. Various properties of the i 's are established. In particular it is shown that i = k i jGj=jGj 2 for some non-negative integers k i and that P 1 i=1 i is either 1 or at most 1/2 for solvable groups.

We prove that fully residually free groups have the Howson property, that is the intersection of any two finitely generated subgroups in such a group is again finitely generated. We also establish some commensurability properties for finitely generated fully residually free groups which are similar to those of free groups. Finally we prove that for a finitely generated fully residually free gro...

We show that for any finitely generated group of matrices that is not virtually solvable, there is an integer m such that, given an arbitrary finite generating set for the group, one may find two elements a and b that are both products of at most m generators, such that a and b are free generators of a free subgroup. This uniformity result improves the original statement of the Tits alternative.