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Geometric dimension of groups for the family of virtually cyclic subgroups
Author(s) -
Degrijse Dieter,
Petrosyan Nansen
Publication year - 2014
Publication title -
journal of topology
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.447
H-Index - 31
eISSN - 1753-8424
pISSN - 1753-8416
DOI - 10.1112/jtopol/jtt045
Subject(s) - mathematics , cyclic group , quotient , dimension (graph theory) , cardinality (data modeling) , torsion (gastropod) , pure mathematics , space (punctuation) , combinatorics , discrete mathematics , computer science , medicine , abelian group , surgery , data mining , operating system
By studying commensurators of virtually cyclic groups, we prove that every elementary amenable group of finite Hirsch length h and cardinality ℵ n admits a finite‐dimensional classifying space with virtually cyclic stabilizers of dimension n + h + 2 . We also provide a criterion for groups that fit into an extension with torsion‐free quotient to admit a finite‐dimensional classifying space with virtually cyclic stabilizers. Finally, we exhibit examples of integral linear groups of type F whose geometric dimension for the family of virtually cyclic subgroups is finite but arbitrarily larger than the geometric dimension for proper actions. This answers a question posed by W. Lück.