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The ratio and generating function of cogrowth coefficients of finitely generated groups

Ryszard Szwarc (1998)

Studia Mathematica

Let G be a group generated by r elements g 1 , … , g r . Among the reduced words in g 1 , … , g r of length n some, say γ n , represent the identity element of the group G. It has been shown in a combinatorial way that the 2nth root of γ 2 n has a limit, called the cogrowth exponent with respect to the generators g 1 , … , g r . We show by analytic methods that the numbers γ n vary regularly, i.e. the ratio γ 2 n + 2 / γ 2 n is also convergent. Moreover, we derive new precise information on the domain of holomorphy of γ(z), the generating function associated...

The rhombic dodecahedron and semisimple actions of Aut(Fₙ) on CAT(0) spaces

Martin R. Bridson (2011)

Fundamenta Mathematicae

We consider actions of automorphism groups of free groups by semisimple isometries on complete CAT(0) spaces. If n ≥ 4 then each of the Nielsen generators of Aut(Fₙ) has a fixed point. If n = 3 then either each of the Nielsen generators has a fixed point, or else they are hyperbolic and each Nielsen-generated ℤ⁴ ⊂ Aut(F₃) leaves invariant an isometrically embedded copy of Euclidean 3-space 𝔼³ ↪ X on which it acts as a discrete group of translations with the rhombic dodecahedron as a Dirichlet...

The Ribes-Zalesskii property of some one relator groups

Gilbert Mantika, Narcisse Temate-Tangang, Daniel Tieudjo (2022)

Archivum Mathematicum

The profinite topology on any abstract group G , is one such that the fundamental system of neighborhoods of the identity is given by all its subgroups of finite index. We say that a group G has the Ribes-Zalesskii property of rank k , or is RZ k with k a natural number, if any product H 1 H 2 ⋯ H k of finitely generated subgroups H 1 , H 2 , ⋯ , H k is closed in the profinite topology on G . And a group is said to have the Ribes-Zalesskii property or is RZ if it is RZ k for any natural number k . In this paper we characterize groups...

The Riemann theorem and divergent permutations

Roman Wituła (1996)

Colloquium Mathematicae

In this paper the fundamental algebraic propeties of convergent and divergent permutations of ℕ are presented. A permutation p of ℕ is said to be divergent if at least one conditionally convergent series ∑ a n of real terms is rearranged by p to a divergent series ∑ a p ( n ) . All other permutations of ℕ are called convergent. Some generalizations of the Riemann theorem about the set of limit points of the partial sums of rearrangements of a given conditionally convergent series are also studied.

The ring of multisymmetric functions

Francesco Vaccarino (2005)

Annales de l’institut Fourier

We give a presentation (in terms of generators and relations) of the ring of multisymmetric functions that holds for any commutative ring R , thereby answering a classical question coming from works of F. Junker [J1, J2, J3] in the late nineteen century and then implicitly in H. Weyl book “The classical groups” [W].

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