We introduce the concepts of degree inertia, diG(H), and compression, dcG(H), a finitely generated subgroup H given group G. For case direct products free-abelian free groups, we compute compression give an upper bound for inertia. Imposing some technical assumptions to supremum involved in definition notion called restricted diG′(H), and, again Zm×Fn, provide explicit formula relating it inert...

Let D be a division algebra such that D ⊗ D is a Noetherian algebra, then any division subalgebra of D is a finitely generated division algebra. Let ∆ be a finite set of commuting derivations or automorphisms of the division algebra D, then the group Ev(∆) of common eigenvalues (i.e. weights) is a finitely generated abelian group. Typical examples of D are the quotient division algebra Frac(D(X...

Let φ : P → P be a morphism of degree d ≥ 2 defined over C. The dynamical Mordell–Lang conjecture says that the intersection of an orbit Oφ(P ) and a subvariety X ⊂ P is usually finite. We consider the number of linear subvarieties L ⊂ P such that the intersection Oφ(P ) ∩ L is “larger than expected.” When φ is the d-power map and the coordinates of P are multiplicatively independent, we prove ...

A long standing open problem in the theory of hyperfinite equivalence relations asks if the orbit equivalence relation generated by a Borel action of a countable amenable group is hyperfinite. In this paper we show that this question has a positive answer when the acting group is locally nilpotent. This extends previous results obtained by Gao–Jackson for abelian groups and by Jackson–Kechris–L...

The Birman exact sequence describes the effect on the mapping class group of a surface with boundary of gluing discs to the boundary components. We construct an analogous exact sequence for the automorphism group of a free group. For the mapping class group, the kernel of the Birman exact sequence is a surface braid group. We prove that in the context of the automorphism group of a free group, ...

Given a smooth geometrically connected curve C over field k and commutative group scheme G of finite type the function K C, we study Tate–Shafarevich groups given by elements $$H^1(K,G)$$ locally trivial at completions associated with closed points C. When comes from k-group is number (or finitely generated has k-point), prove finiteness generalizing result Saïdi Tamagawa for abelian varieties....

a subgroup h of a group g is called inert if‎, ‎for each $gin g$‎, ‎the index of $hcap h^g$ in $h$ is finite‎. ‎we give a classification ‎of soluble-by-finite groups $g$ in which subnormal subgroups are inert in the cases where $g$ has no nontrivial torsion normal subgroups or $g$‎ ‎is finitely generated‎.

Schur's theorem and its generalisation, Baer's theorem, are distinguished results in group theory, connecting the upper central quotients with lower series. The aim of this paper is to generalise these two different directions, using novel methods related non-abelian tensor product. In particular, we prove a version Schur–Baer for finitely generated groups. Then, apply newly obtained describe k...

The p-adic Simpson correspondence due to Faltings (Adv Math 198(2):847–862, 2005) is a analogue of non-abelian Hodge theory. following the main result this article: for line bundles can be enhanced rigid analytic morphism moduli spaces under certain smallness conditions. In complex setting, shows that there from space vector with integrable connection representations finitely generated group as...

Finitely generated Z-modules have canonical decompositions. When such modules are given in a finitely presented form there is a classical algorithm for computing a canonical decomposition. This is the algorithm for computing the Smith normal form of an integer matrix. We discuss algorithms for Smith normal form computation, and present practical algorithms which give excellent performance for m...