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. ${\displaystyle n}$

Growth of graphs

Let it be a graph and a firmly chosen node . ${\displaystyle (V,E)}$ ${\displaystyle v_{0}\in V}$

For is the number of nodes for which there is a path of at most edges from to . ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle a_{n}}$ ${\displaystyle v\in V}$ ${\displaystyle n}$ ${\displaystyle v_{0}}$ ${\displaystyle v}$

The growth rate of the graph is by definition the growth rate of the sequence . ${\displaystyle (a_{n})_{n}}$

Growth of groups

Let it be a finitely generated group and a finite generating system . The growth rate of the Cayley graph for . ${\displaystyle G}$ ${\displaystyle S}$ ${\displaystyle G}$ ${\displaystyle S}$

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 . ${\displaystyle S=\{s_{1},\dots ,s_{k}\}}$ ${\displaystyle g\in G}$ ${\displaystyle g=s_{i_{1}}^{e_{1}}\cdots s_{i_{m}}^{e_{m}}}$ ${\displaystyle m\in \mathbb {N} }$ ${\displaystyle i_{1},\dots ,i_{m}}$ ${\displaystyle \{1,\dots ,n\}}$ ${\displaystyle e_{1},\dots ,e_{m}\in \mathbb {Z} }$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle a_{n}}$ ${\displaystyle G}$ ${\displaystyle |e_{1}|+\dots +|e_{m}|\leq n}$ ${\displaystyle G}$ ${\displaystyle (a_{n})_{n}}$

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. ${\displaystyle (a_{n})_{n}}$ ${\displaystyle G}$

Examples

• The growth of is linear. ${\displaystyle \mathbb {Z} }$
• The growth of is square. ${\displaystyle \mathbb {Z} ^{2}}$
• The growth of a nilpotent group is polynomial in degree , with the Abelian groups in the descending central series of and being their rank . ${\displaystyle \sum rk(L_{i}(G))}$ ${\displaystyle L_{i}(G)}$ ${\displaystyle G}$ ${\displaystyle rk(L_{i}(G))}$
• 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

1. ^ M. Gromow: Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. No. 53: 53-73 (1981).
2. ↑ B. Kleiner: A new proof of Gromov's theorem on groups of polynomial growth. J. Amer. Math. Soc. 23 (2010), no. 3, 815-829.
3. ^ 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.
4. ^ J. Milnor: A note on curvature and fundamental group. J. Differential Geometry, 2: 1-7 (1968).