Area group

In group theory , a branch of mathematics that are fundamental groups of closed orientable surfaces as surface groups (ger .: surface groups ), respectively.

definition

Let be a natural number and the closed , orientable area of gender . ${\ displaystyle g \ geq 1}$ ${\ displaystyle S_ {g}}$ ${\ displaystyle g}$

${\ displaystyle g = 1}$ : Torus
${\ displaystyle g = 2}$ : Pretzel surface
${\ displaystyle g = 3}$

The fundamental groups are referred to as surface groups . ${\ displaystyle \ pi _ {1} S_ {g}}$

Presentation

The area group has the presentation ${\ displaystyle \ pi _ {1} S_ {g}}$

{\ 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}

.

For example is . ${\ displaystyle \ pi _ {1} S_ {1} = \ mathbb {Z} ^ {2}}$

Hyperbolicity

With the exception of , all face groups are hyperbolic . Max Dehn used hyperbolic geometry to solve the word problem for groups of surfaces. This work is considered to be the forerunner of the theory of hyperbolic groups developed by Gromow in the 1980s . ${\ displaystyle \ pi _ {1} S_ {1} = \ mathbb {Z} ^ {2}}$

Area groups are - like all hyperbolic groups - automatic groups , so their word problem can be solved in quadratic time.

Representations (higher pond mill theory)

The theory of the representations of surface groups in Lie groups is called the higher Teichmüller theory . Classic Teichmüller theory is the special case , in this case the holonomy mediates a bijection between the Teichmüller space and a coherent component of . ${\ displaystyle \ pi _ {1} S_ {g}}$ ${\ displaystyle G}$ ${\ displaystyle G = PSL (2, \ mathbb {R})}$ ${\ displaystyle Rep (\ pi _ {1} S_ {g}, G): = Hom (\ pi _ {1} S_ {g}, PSL (2, \ mathbb {R})) / conj.}$

Connected components of the representation variety

In the following, denotes the representation variety whose connected components - for connected Lie groups - correspond to the connected components of. ${\ displaystyle Hom (\ pi _ {1} S_ {g}, G)}$ ${\ displaystyle G}$ ${\ displaystyle Rep (\ pi _ {1} S_ {g})}$

For compact, connected groups , the connected components of the representation variety correspond to the elements of . ${\ displaystyle G}$ ${\ displaystyle \ pi _ {1} G}$

For , the related components of the representation variety are classified by the values of the Euler class . Because, according to the Milnor-Wood inequality, the Euler class can take exactly the integer values in the interval , the representation variety has connected components. A representation is faithful with a discrete image exactly when . ${\ displaystyle G = PSL (2, \ mathbb {R})}$ ${\ displaystyle e}$ ${\ displaystyle \ left [\ chi (S_ {g}), - \ chi (S_ {g}) \ right]}$ ${\ displaystyle 4g-3}$ ${\ displaystyle \ mid e \ mid = \ chi (S_ {g})}$

For the representation variety has contextual components.

{\ displaystyle G = SL (2, \ mathbb {R})}

{\ displaystyle 2 ^ {2g + 1} + 2g-3}

For or the connected components of are classified by the values of the second Stiefel-Whitney class , the representation variety has two connected components . ${\ displaystyle G = PSL (2, \ mathbb {C})}$ ${\ displaystyle G = SO (3)}$ ${\ displaystyle Rep (\ pi _ {1} S_ {g}, G)}$ ${\ displaystyle w_ {2}}$

For or , the representation variety is coherent.

{\ displaystyle G = SL (2, \ mathbb {C})}

{\ displaystyle G = SU (2)}

For with , the representation variety has 3 components if is odd and 6 components if is even. The proof uses the theory of the Higgs bundle . ${\ displaystyle G = PSL (n, \ mathbb {R})}$ ${\ displaystyle n \ geq 3}$ ${\ displaystyle n}$ ${\ displaystyle n}$

literature

Heiner Zieschang , Elmar Vogt , Hans-Dieter Coldewey: Areas and flat discontinuous groups (= Lecture Notes in Mathematics. Vol. 122). Springer, Berlin et al. 1970.

credentials

^ Max Dehn : About infinite discontinuous groups. In: Mathematical Annals . Vol. 71, 1912, pp. 116-144 .
↑ Michael F. Atiyah , Raoul Bott : The Yang-Mills equations over Riemann surfaces. In: Philosophical Transactions of the Royal Society . Series A: Mathematical, Physical and Engineering Sciences. Vol. 305, No. 1505, 1983, pp. 523-615, doi : 10.1098 / rsta.1983.0017 .
^ William Mark Goldman : Discontinuous groups and the Euler class. University of California, Berkeley CA 1980 (Thesis (Ph. D. in Mathematics)).
^ Nigel J. Hitchin : Lie Groups and Teichmüller space. In: Topology. Vol. 31, No. 3, 1992, 449-473, doi : 10.1016 / 0040-9383 (92) 90044-I .

[1] Max Dehn : About infinite discontinuous groups. In: Mathematical Annals . Vol. 71, 1912, pp. 116-144 .

[2] Michael F. Atiyah , Raoul Bott : The Yang-Mills equations over Riemann surfaces. In: Philosophical Transactions of the Royal Society . Series A: Mathematical, Physical and Engineering Sciences. Vol. 305, No. 1505, 1983, pp. 523-615, doi : 10.1098 / rsta.1983.0017 .

[3] William Mark Goldman : Discontinuous groups and the Euler class. University of California, Berkeley CA 1980 (Thesis (Ph. D. in Mathematics)).

[4] Nigel J. Hitchin : Lie Groups and Teichmüller space. In: Topology. Vol. 31, No. 3, 1992, 449-473, doi : 10.1016 / 0040-9383 (92) 90044-I .