Growth (group theory)
In the mathematical field of group theory, the rate of growth of a group roughly counts the number of elements that can be represented as the products of length from given producers.
Growth of graphs
Let it be a graph and a firmly chosen node .
For is the number of nodes for which there is a path of at most edges from to .
The growth rate of the graph is by definition the growth rate of the sequence .
Growth of groups
Let it be a finitely generated group and a finite generating system . The growth rate of the Cayley graph for .
More precisely, this means the following: If , then each group element can be written as a word , where , the indices are elements of and the exponents are any whole numbers. For each, let the number of elements of that have such a spelling . The growth rate of the group is then precisely the growth rate of the consequence .
Different generating systems give different Cayley graphs and thus also different sequences , but the Cayley graphs of different finite generating systems are bilipschitz-equivalent to each other , so that the growth rate of the sequence only depends on the group and not on the selected generating system.
Examples
- The growth of is linear.
- The growth of is square.
- The growth of a nilpotent group is polynomial in degree , with the Abelian groups in the descending central series of and being their rank .
- Gromow's theorem: A group has polynomial growth if and only if it is virtually nilpotent .
- Milnor-Wolf Theorem: A solvable group has either polynomial or exponential growth.
- The Grigorchuk group has sub-exponential but not polynomial growth.
- The growth of a nonabelian free group is exponential.
- Fundamental groups of compact Riemannian manifolds of negative section curvature have exponential growth.
literature
- J. Milnor: Growth of finitely generated solvable groups. J. Differential Geometry, 2: 447-449 (1968).
Web links
- M. Duchin: Counting in Groups: Fine Asymptotic Geometry , Notices of the American Mathematical Society , September 2016
Individual evidence
- ^ M. Gromow: Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. No. 53: 53-73 (1981).
- ↑ B. Kleiner: A new proof of Gromov's theorem on groups of polynomial growth. J. Amer. Math. Soc. 23 (2010), no. 3, 815-829.
- ^ RI Grigortschuk: Degrees of growth of finitely generated groups and the theory of invariant means. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984) no. 5, 939-985.
- ^ J. Milnor: A note on curvature and fundamental group. J. Differential Geometry, 2: 1-7 (1968).