Virtual property
In mathematics , a group is said to have a virtual property if this property applies to a subgroup of finite index . One speaks, for example, of virtually Abelian , virtually nilpotent or virtually cyclical groups.
A prominent example of a virtual property is the virtual hook conjecture , proven by Ian Agol in 2012 , for which he received the $ 3 million Breakthrough Prize in Mathematics in 2016 .
definition
Let P be a property of groups. Then it is said that a group virtually P is when a subgroup of finite index has the property P has.
About the correspondence between overlays and subgroups of the fundamental group of this speech can be on manifolds transfer: It is said that a manifold is virtually P if there is a finite interference with property P there.
Examples
- A group is virtual cyclic if and only if it is a semidirect product of a cyclic normal divisor and a finite group.
- Every virtual cyclic group is a semidirect product of a finite normal divisor and either or .
- Semi direct products of a finite and an abelian group (or vice versa) are virtually abelian . For example, generalized dihedral groups are virtually Abelian.
- Semi-direct products from a finite and a nilpotent group (or vice versa) are virtually nilpotent .
- Gromow's theorem: A group is virtually nilpotent if and only if it has polynomial growth .
- Free products of finite groups are virtually free . For example, the module group is virtually free.
- Agol's theorem: Every compact , orientable , irreducible 3-manifold is virtually hooked , virtually frayed and has a virtually positive first Betti number .
literature
- John Stallings: Groups of dimension 1 are locally free. Bull. Amer. Math. Soc. 74 1968 361-364.
- Michael Gromow: Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. No. 53: 53-73 (1981).
- Thomas Farrell, Lowell Jones: The lower algebraic K-theory of virtually infinite cyclic groups. K-Theory 9 (1995), no. 1, 13-30.
- Daniel Juan-Pineda, Ian Leary: On classifying spaces for the family of virtually cyclic subgroups. Recent developments in algebraic topology, 135-145, Contemp. Math., 407, Amer. Math. Soc., Providence, RI, 2006.
- Wolfgang Lück: Survey on classifying spaces for families of subgroups. Infinite groups: geometric, combinatorial and dynamical aspects, 269–322, Progr. Math., 248, Birkhäuser, Basel, 2005.
- Ian Agol: The virtual hook conjecture. With an appendix by Agol, Daniel Groves, and Jason Manning. Doc. Math. 18 (2013), 1045-1087.
Web links
- Ian Agol: Virtual Properties of 3-Manifolds