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Some questions about the dimension of a group action
Author(s) -
Bleak Collin
Publication year - 2008
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdn060
Subject(s) - embedding , mathematics , dimension (graph theory) , mathematical proof , group (periodic table) , piecewise linear function , pure mathematics , action (physics) , discrete mathematics , combinatorics , mathematical analysis , computer science , geometry , quantum mechanics , artificial intelligence , physics
This article discusses three families of groups: Z ≀ Z n , PL( I n ), and PL( S n ) (the last two being the families of groups of piecewise‐linear homeomorphisms of standard n ‐dimensional spaces). It is shown that for positive n ∈ N , Z ≀ Z n embeds in PL( I n ), which embeds in PL( S n ). It is known that Z ≀ Z 2 fails to embed in PL( I 1 ), and this article extends that previous result to show that Z ≀ Z 2 also fails to embed in PL( S 1 ). The nature of the proofs of these embedding and non‐embedding results hints that there may be corresponding non‐embedding results in higher dimensions.