Nielsen transformation

In mathematics , Nielsen transformations are an important tool in combinatorial group theory ; they are named after the mathematician Jakob Nielsen .

definition

Let be a group and an ordered n-tuple of elements . An elementary Nielsen transform is one of the following three types of substitution: ${\displaystyle G}$ ${\displaystyle (g_{1},\ldots ,g_{n})}$ ${\displaystyle G}$

• For a replace by . ${\displaystyle i\in \left\{1,\ldots ,n\right\}}$ ${\displaystyle g_{i}}$ ${\displaystyle g_{i}^{-1}}$
• For two, swap and . ${\displaystyle i\not =j\in \left\{1,\ldots ,n\right\}}$ ${\displaystyle g_{i}}$ ${\displaystyle g_{j}}$
• For two replace by . ${\displaystyle i\not =j\in \left\{1,\ldots ,n\right\}}$ ${\displaystyle g_{i}}$ ${\displaystyle g_{i}g_{j}}$

A Nielsen transformation is a sequence of finitely many elementary Nielsen transformations. Two ordered tuples are called Nielsen equivalent if they emerge from each other through a Nielsen transformation.

Applications

Generating systems of free groups

Be the free group with producers . Then every minimal generating system has elements and a -tuple is a generating system of if and only if the ordered tuples and are Nielsen-equivalent. ${\displaystyle F_{n}}$ ${\displaystyle n}$ ${\displaystyle x_{1},\ldots ,x_{n}}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle g_{1},\ldots ,g_{n}}$ ${\displaystyle F_{n}}$ ${\displaystyle (x_{1},\ldots ,x_{n})}$ ${\displaystyle (g_{1},\ldots ,g_{n})}$

Generating systems of surface groups

Let the face group be of the gender . Then every minimal generating system has elements and a -tuple is a generating system of if and only if the ordered tuples and are Nielsen-equivalent. ${\displaystyle \pi _{1}S_{g}=\langle a_{1},b_{1},\ldots ,a_{g},b_{g}\mid \Pi _{i=1}^{g}\left[a_{i},b_{i}\right]=1\rangle }$ ${\displaystyle g}$ ${\displaystyle 2g}$ ${\displaystyle 2g}$ ${\displaystyle h_{1},\ldots ,h_{2g}}$ ${\displaystyle \pi _{1}S_{g}}$ ${\displaystyle (a_{1},b_{1},\ldots ,a_{g},b_{g})}$ ${\displaystyle (h_{1},\ldots ,h_{2g})}$

literature

1. ↑ Jakob Nielsen : About the isomorphisms of infinite groups without relation . Math. Ann. 79 (1918), no.3, 269-272. doi : 10.1007 / BF01458209
2. ↑ Jakob Nielsen: Om regning med ikke-kommutative factors og dens anvendelse i gruppeteorien . Math. Tidsskrift B (1921), 78-94.
3. ↑ Heiner Zieschang : About the Nielsen method of shortening in free products with amalgam . Invent. Math. 10: 4-37 (1970). doi : 10.1007 / BF01402968