finite conjugacy
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2021 ◽  
pp. 1-46
Author(s):  
YOSHIKATA KIDA ◽  
ROBIN TUCKER-DROB

Abstract We show that every countable group with infinite finite conjugacy (FC)-center has the Schmidt property, that is, admits a free, ergodic, measure-preserving action on a standard probability space such that the full group of the associated orbit equivalence relation contains a non-trivial central sequence. As a consequence, every countable, inner amenable group with property (T) has the Schmidt property.



2018 ◽  
Vol 105 (1) ◽  
pp. 24-33
Author(s):  
M. DE FALCO ◽  
F. DE GIOVANNI ◽  
C. MUSELLA ◽  
N. TRABELSI

If $k$ is a positive integer, a group $G$ is said to have the $FE_{k}$-property if for each element $g$ of $G$ there exists a normal subgroup of finite index $X(g)$ such that the subgroup $\langle g,x\rangle$ is nilpotent of class at most $k$ for all $x\in X(g)$. Thus, $FE_{1}$-groups are precisely those groups with finite conjugacy classes ($FC$-groups) and the aim of this paper is to extend properties of $FC$-groups to the case of groups with the $FE_{k}$-property for $k>1$. The class of $FE_{k}$-groups contains the relevant subclass $FE_{k}^{\ast }$, consisting of all groups $G$ for which to every element $g$ there corresponds a normal subgroup of finite index $Y(g)$ such that $\langle g,U\rangle$ is nilpotent of class at most $k$, whenever $U$ is a nilpotent subgroup of class at most $k$ of $Y(g)$.



2018 ◽  
Vol 69 (3) ◽  
pp. 1047-1051 ◽  
Author(s):  
Gláucia Dierings ◽  
Pavel Shumyatsky


2017 ◽  
Vol 09 (04) ◽  
pp. 707-716
Author(s):  
Sherry Gong

In this paper, we show that if [Formula: see text] is a non-elementary word hyperbolic group, [Formula: see text] is an element, and the conjugacy class of [Formula: see text] is infinite, then all traces [Formula: see text] vanish on [Formula: see text]. Moreover, we completely classify all traces by showing that traces [Formula: see text] are linear combinations of traces [Formula: see text] given by [Formula: see text] where [Formula: see text] is an element with finite conjugacy class, denoted [Formula: see text]. We demonstrate these two statements by introducing a new method to study traces that uses Sobolev norms and the rapid decay property.



2017 ◽  
Vol 96 (3) ◽  
pp. 429-437 ◽  
Author(s):  
COSTANTINO DELIZIA ◽  
PAVEL SHUMYATSKY ◽  
ANTONIO TORTORA

Let $w$ be a group-word. For a group $G$, let $G_{w}$ denote the set of all $w$-values in $G$ and let $w(G)$ denote the verbal subgroup of $G$ corresponding to $w$. The group $G$ is an $FC(w)$-group if the set of conjugates $x^{G_{w}}$ is finite for all $x\in G$. It is known that if $w$ is a concise word, then $G$ is an $FC(w)$-group if and only if $w(G)$ is $FC$-embedded in $G$, that is, the conjugacy class $x^{w(G)}$ is finite for all $x\in G$. There are examples showing that this is no longer true if $w$ is not concise. In the present paper, for an arbitrary word $w$, we show that if $G$ is an $FC(w)$-group, then the commutator subgroup $w(G)^{\prime }$ is $FC$-embedded in $G$. We also establish the analogous result for $BFC(w)$-groups, that is, groups in which the sets $x^{G_{w}}$ are boundedly finite.



2017 ◽  
Vol 67 (2) ◽  
Author(s):  
Francesco de Giovanni ◽  
Federica Saccomanno

AbstractA group



2016 ◽  
Vol 15 (05) ◽  
pp. 1650093
Author(s):  
Alex Carrazedo Dantas

Let [Formula: see text] be the group of automorphisms of the one-rooted [Formula: see text]-ary tree and [Formula: see text] be a transitive state-closed subgroup of [Formula: see text] with bounded finite conjugacy classes. We prove that the torsion subgroup Tor[Formula: see text] has finite exponent and determine an upper bound for the exponent. In case [Formula: see text] is a prime number, we prove that [Formula: see text] is either a torsion group or a torsion-free abelian group.



2015 ◽  
Vol 431 ◽  
pp. 24-37
Author(s):  
M. De Falco ◽  
F. de Giovanni ◽  
C. Musella ◽  
N. Trabelsi


2013 ◽  
Vol 89 (1) ◽  
pp. 41-48 ◽  
Author(s):  
M. DE FALCO ◽  
F. DE GIOVANNI ◽  
C. MUSELLA ◽  
N. TRABELSI

AbstractA group $G$ is said to be an $FC$-group if each element of $G$ has only finitely many conjugates, and $G$ is minimal non$FC$ if all its proper subgroups have the property $FC$ but $G$ is not an $FC$-group. It is an open question whether there exists a group of infinite rank which is minimal non$FC$. We consider here groups of infinite rank in which all proper subgroups of infinite rank are $FC$, and prove that in most cases such groups are either $FC$-groups or minimal non$FC$.



2012 ◽  
Vol 19 (spec01) ◽  
pp. 1197-1204
Author(s):  
Maria De Falco ◽  
Francesco de Giovanni ◽  
Carmela Musella

The structure of groups for which certain sets of commutator subgroups are finite is investigated. In particular, the relation between such groups and groups with finite conjugacy classes of elements is discussed.



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