Geometric group theory

The geometric group theory is that part of group theory that pays special attention to the interaction between geometric objects and the groups operating on them . It is particularly about group operations on graphs and metric spaces , ultimately the groups themselves become such geometric objects.

Group operations

If there is a category and if this category is an object, then the set of automorphisms is a group. Every homomorphism of a group in this automorphism group is then called a representation or operation of on . If, for example, the category is vector spaces with linear mappings , then one obtains the classical representation theory of groups , in which, after choosing a vector space basis, each group element is mapped onto a regular matrix . If the category of all sets is nothing other than the group of all permutations on the set . These two ways of looking at things, matrix groups and permutation groups, lay at the beginning of group theory. ${\displaystyle {\mathcal {C}}}$ ${\displaystyle X}$ ${\displaystyle \mathrm {Aut} _{\mathcal {C}}(X)}$ ${\displaystyle X\rightarrow X}$ ${\displaystyle G\rightarrow \mathrm {Aut} _{\mathcal {C}}(X)}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \mathrm {Aut} _{\mathcal {C}}(X)}$ ${\displaystyle X}$

In geometric group theory, categories are used instead, the objects of which have a more geometric character, namely graphs and metric spaces with suitable morphisms . The automorphism groups have been used for a long time to investigate the symmetry properties of objects. Conversely, however, group properties can be studied through their operations on objects and groups themselves can be made into geometric objects, so that geometric concept formations become meaningful for these groups.

The Cayley graph named after Arthur Cayley is assigned to each group and a generating system of this group . The nodes of these graphs are the group elements themselves and every two nodes are connected by an edge if one of the nodes is the product of the other and an element of the generating system. The group operates on this graph by multiplying from the left, because the nodes are themselves group elements. Group properties carry over to properties of operations on graphs. For example, free groups can be characterized by the fact that they operate freely on a tree . Since the latter is evidently carried over to subgroups , one obtains an elegant proof of the Nielsen-Schreier theorem , according to which every subgroup of a free group is free again. This is a purely algebraic theorem, which also has purely algebraic proofs, but whose geometrically motivated proof indicated here is more easily accessible. This is considered a standard application of geometric group theory.

Quasi-isometry

As a rule, one restricts oneself to finitely generated groups, because only for finite generating systems one obtains a Cayley graph in which only finitely many edges start from each node. In a further step one considers finitely generated groups by means of their Cayley graphs as metric spaces with the path length between two nodes as the distance. As the graph contiguous is to always find ways to finite length. If you replace each edge with an isometric image of the unit interval without overlapping , you even get a geodetic metric space that contains the nodes of the caley graph as a subspace. On the class of such spaces one considers equivalence classes of quasi-isometries as morphisms , whereby two quasi-isometries are called equivalent if they have a limited distance. Here the Švarc-Milnor theorem plays an important role, which establishes quasi-isometrics between groups and the metric spaces on which they operate. Now one can even speak of the quasi-isometry of two groups, namely as the quasi-isometry of the associated metric spaces, and there is no longer any dependence on the selected finite generating system with regard to the quasi-isometry. For example, the group is quasi-isometric to the metric space , but not to the group , because the latter is not even finite. It is one of the main goals of geometric group theory to understand the classification of the finitely generated groups with respect to quasi-isometry. ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$

The simplest and at the same time trivial case is that of the finite groups , because these are characterized by Cayley graphs of finite diameter and therefore form a single quasi-isometric class. The geometric group theory is therefore trivial for finite groups. The next simple case is the quasi-isometric class of , which consists of precisely those groups that contain a subgroup that is too isomorphic with a finite index . This includes groups of the species or the infinite dihedral group . ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Z} \times \mathbb {Z} /(n)}$

There are uncountably many quasi-isometric classes of finitely generated groups and a complete classification seems to be a long way off. In order to get closer to a possible classification down to quasi-isometry, one is interested in properties that are invariant under quasi-isometry; such invariant properties are called geometric. Many far-reaching results are obtained from algebraic characterizations of geometric properties. One important example is the growth of groups . Quasi-isometric groups belong to the same growth class, that is, the growth class is a geometric property, and according to Gromov's theorem a group has polynomial growth if and only if it has a nilpotent subgroup with a finite index. Other important geometric properties are, for example, indirectness or hyperbolicity of groups. The latter in particular is a convincing example of how purely geometrical concept formations in geometrical group theory become group properties. These then have purely algebraic consequences, for example the word problem for hyperbolic groups can be solved or every infinite hyperbolic group has an element of infinite order . Conversely, the results of geometric group theory contribute to the understanding of geometric objects, for example the fundamental group of a compact , coherent Riemannian manifold is finitely generated without a boundary, operates on the universal superposition by means of cover transformations , and this operation represents a quasi-isometry between the fundamental group and the universal overlay as metric space. Group properties of the fundamental groups have consequences for the geometry of the Riemannian manifolds.

Individual evidence

1. ^ Stephan Rosebrock: Geometrical group theory , Vieweg-Verlag 2004, ISBN 3-528-03212-X
2. Jump up ↑ Pierre de la Harpe: Topics in Geometric Group Theory , University Of Chicago Press 2000, ISBN 0-226-31721-8
3. ↑ Clara Löh: Geometric Group Theory , Springer-Verlag 2017, ISBN 978-3-319-72253-5 , page 127
4. ↑ Clara Löh: Geometric Group Theory , Springer-Verlag 2017, ISBN 978-3-319-72253-5 , definition 5.6.6
5. ^ Clara Löh: Geometric Group Theory , Springer-Verlag 2017, ISBN 978-3-319-72253-5 , Corollary 5.4.10