scholarly journals The Martin boundary of a free product of abelian groups

2020 ◽  
Vol 70 (1) ◽  
pp. 313-373
Author(s):  
Matthieu Dussaule
Keyword(s):  
Free Product ◽  
Abelian Groups ◽  
Martin Boundary

Construction of an infinitely generated group that is not a free product of surface groups and abelian groups, but which acts freely on an ℝ-tree

1998 ◽  
Vol 128 (2) ◽  
pp. 433-445 ◽  
Author(s):  
Andeas Zastrow
Keyword(s):  
Fundamental Group ◽  
Free Product ◽  
Abelian Groups ◽  
Length Function ◽  
Surface Groups ◽  
Finitely Generated ◽  
Combinatorial Description ◽  
Finitely Generated Groups ◽  
Special Subgroup

The existence of a group H as described in the title shows that the statement of Rips's Theorem for finitely generated groups cannot be extended without further complications to infinitely generated groups. The construction as given in this paper uses a careful combinatorial description of the fundamental group of the Hawaiian Earrings and a length function that can be put on a special subgroup. Then the existence of H follows using a theorem of Chiswell, Alperin and Moss.

scholarly journals Relative hyperbolicity for automorphisms of free products and free groups

2020 ◽  
pp. 1-38
Author(s):  
François Dahmani ◽  
Ruoyu Li
Keyword(s):  
Free Product ◽  
Free Group ◽  
Semidirect Product ◽  
Abelian Groups ◽  
Free Groups ◽  
Conjugacy Problem ◽  
Product Formula ◽  
Free Products ◽  
Relatively Hyperbolic ◽  
Relative Hyperbolicity

We prove that for a free product [Formula: see text] with free factor system [Formula: see text], any automorphism [Formula: see text] preserving [Formula: see text], atoroidal (in a sense relative to [Formula: see text]) and none of whose power send two different conjugates of subgroups in [Formula: see text] on conjugates of themselves by the same element, gives rise to a semidirect product [Formula: see text] that is relatively hyperbolic with respect to suspensions of groups in [Formula: see text]. We recover a theorem of Gautero–Lustig and Ghosh that, if [Formula: see text] is a free group, [Formula: see text] an automorphism of [Formula: see text], and [Formula: see text] is its family of polynomially growing subgroups, then the semidirect product by [Formula: see text] is relatively hyperbolic with respect to the suspensions of these subgroups. We apply the first result to the conjugacy problem for certain automorphisms (atoroidal and toral) of free products of abelian groups.

Additive bases via Fourier analysis

2021 ◽  
pp. 1-12
Author(s):  
Bodan Arsovski
Keyword(s):  
Abelian Group ◽  
Fourier Analysis ◽  
Vector Space ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Probabilistic Method ◽  
Additive Basis ◽  
Generating Sets

Abstract Extending a result by Alon, Linial, and Meshulam to abelian groups, we prove that if G is a finite abelian group of exponent m and S is a sequence of elements of G such that any subsequence of S consisting of at least $$|S| - m\ln |G|$$ elements generates G, then S is an additive basis of G . We also prove that the additive span of any l generating sets of G contains a coset of a subgroup of size at least $$|G{|^{1 - c{ \in ^l}}}$$ for certain c=c(m) and $$ \in = \in (m) < 1$$ ; we use the probabilistic method to give sharper values of c(m) and $$ \in (m)$$ in the case when G is a vector space; and we give new proofs of related known results.

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