15. Lamplighter Groups

2017 ◽  
pp. 310-330
Keyword(s):  
Lamplighter Groups

scholarly journals Automorphisms of higher rank lamplighter groups

2015 ◽  
Vol 25 (08) ◽  
pp. 1275-1299 ◽  
Author(s):  
Melanie Stein ◽  
Jennifer Taback ◽  
Peter Wong
Keyword(s):  
Cayley Graph ◽  
Infinite Number ◽  
Conjugacy Classes ◽  
Generating Set ◽  
Twisted Conjugacy ◽  
Higher Rank ◽  
Lamplighter Groups

Let [Formula: see text] denote the group whose Cayley graph with respect to a particular generating set is the Diestel–Leader graph [Formula: see text], as described by Bartholdi, Neuhauser and Woess. We compute both [Formula: see text] and [Formula: see text] for [Formula: see text], and apply our results to count twisted conjugacy classes in these groups when [Formula: see text]. Specifically, we show that when [Formula: see text], the groups [Formula: see text] have property [Formula: see text], that is, every automorphism has an infinite number of twisted conjugacy classes. In contrast, when [Formula: see text] the lamplighter groups [Formula: see text] have property [Formula: see text] if and only if [Formula: see text].

scholarly journals On the spectrum of lamplighter groups and percolation clusters

2008 ◽  
Vol 342 (1) ◽  
pp. 69-89 ◽  
Author(s):  
Franz Lehner ◽  
Markus Neuhauser ◽  
Wolfgang Woess
Keyword(s):  
Percolation Clusters ◽  
Lamplighter Groups

scholarly journals Proper actions of lamplighter groups associated with free groups

2008 ◽  
Vol 346 (3-4) ◽  
pp. 173-176 ◽  
Author(s):  
Yves de Cornulier ◽  
Yves Stalder ◽  
Alain Valette
Keyword(s):  
Free Groups ◽  
Proper Actions ◽  
Lamplighter Groups

scholarly journals Dead end words in lamplighter groups and other wreath products

2005 ◽  
Vol 56 (2) ◽  
pp. 165-178 ◽  
Author(s):  
Sean Cleary† ◽  
Jennifer Taback‡
Keyword(s):  
Wreath Products ◽  
Dead End ◽  
Lamplighter Groups

scholarly journals A characterization of superreflexivity through embeddings of lamplighter groups

2019 ◽  
Vol 147 (11) ◽  
pp. 4745-4755 ◽  
Author(s):  
Mikhail I. Ostrovskii ◽  
Beata Randrianantoanina
Keyword(s):  
Lamplighter Groups

scholarly journals Hyperbolic structures on wreath products

2020 ◽  
Vol 23 (2) ◽  
pp. 357-383
Author(s):  
Sahana H. Balasubramanya
Keyword(s):  
Wreath Products ◽  
Hyperbolic Structures ◽  
Open Questions ◽  
Lamplighter Groups

AbstractThe study of the poset of hyperbolic structures on a group G was initiated by C. Abbott, S. Balasubramanya and D. Osin, [Hyperbolic structures on groups, Algebr. Geom. Topol. 19 2019, 4, 1747–1835, 10.2140/agt.2019.19.1747]. However, this poset is still very far from being understood, and several questions remain unanswered. In this paper, we give a complete description of the poset of hyperbolic structures on the lamplighter groups {\mathbb{Z}_{n}\mathbin{\mathrm{wr}}\mathbb{Z}} and obtain some partial results about more general wreath products. As a consequence of this result, we answer two open questions regarding quasi-parabolic structures: we give an example of a group G with an uncountable chain of quasi-parabolic structures and prove that the lamplighter groups {\mathbb{Z}_{n}\mathbin{\mathrm{wr}}\mathbb{Z}} all have finitely many quasi-parabolic structures.

scholarly journals K -homology and K -theory for the lamplighter groups of finite groups

2017 ◽  
Vol 115 (6) ◽  
pp. 1207-1226 ◽  
Author(s):  
Ramón Flores ◽  
Sanaz Pooya ◽  
Alain Valette
Keyword(s):  
Finite Groups ◽  
Lamplighter Groups ◽  
K Theory

scholarly journals The degree of commutativity and lamplighter groups

2018 ◽  
Vol 28 (07) ◽  
pp. 1163-1173 ◽  
Author(s):  
Charles Garnet Cox
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Wreath Products ◽  
Infinite Groups ◽  
Lamplighter Groups

The degree of commutativity of a group [Formula: see text] measures the probability of choosing two elements in [Formula: see text] which commute. There are many results studying this for finite groups. In [Y. Antolín, A. Martino and E. Ventura, Degree of commutativity of infinite groups, Proc. Amer. Math. Soc. 145 (2017) 479–485, MR 3577854], this was generalized to infinite groups. In this note, we compute the degree of commutativity for wreath products of the form [Formula: see text] and [Formula: see text], where [Formula: see text] is any finite group.

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