Sym metry , I n tegra b i l i ty a nd Geometry : Methods a nd App l i ca t i ons S I GMA 6 ( 20 1 0 ) , 76 , 45 pa ges Erlangen Program at Large - 1 : Geometry of Invariants Vladimir V . KISIL School of Mathematics , University of Leeds , Leeds LS 2 9 JT , UK E - mail : k is i l v @ maths.leeds. ac.uk URL : h t tp : / / www . mat hs . l e eds . ac . uk = ∼ k i s i lv / Received April 20 , 201 0 , in final form September 1 0 , 20 1 0 ; Published online September 26 , 20 1 0 doi:10.3842/SIG MA .2010.076 Abstract . This paper presents geometrical foundation for a systematic treatment of three main ( elliptic , parabolic and hyperbolic ) types of analytic function theories based on the representation theory of SL2(R) group . We describe here geometries of corresponding do - mains . The principal ro ^ le is played by Clifford algebras of matching types . In this paper we also generalise the Fillmore { Springer { Cnops construction which describes cycles as points in the extended space . This allows to consider many algebraic and geometric invariants of cycles within the Erlangen program approach . Key words : analytic function theory ; semisimple groups ; elliptic ; parabolic ; hyperbolic ; Clifford algebras ; complex numbers ; dual numbers ; double numbers ; split - complex numbers ; Mo ¨ bius transformations 201 0 Mathematics Subject Classification : 30 G 35 ; 22 E 46 ; 30 F 45 ; 32 F 45 Contents 1 I n t r o d u ct io n 2 1 . 1 Backgr o u n d a n d h i s t o r y . ..................... 2 1 . 2 H igh l ight s o f o b t ai n ed r e s u l t s . ..................... 3 1 . 3 T he p a p e r o utlin e . ..................... 4 2 E l l ipt ic , pa r abo l ic and hyp erbolic h o m ogen eous spa c e s 5 2:1 SL2( R) gr o up a n d Cli f for d alg e br as . ................5 2 . 2 A ctio n s o f s u b gr o u p s . ..................... 6 2 . 3 I n v a r ian c e o f c y c l e s . ..................... 1 0 3 Space of cy cle s 1 1 3 . 1 F i l l m o r e { Sprin g e r { C n o p s c o n s t r u c t i o n ( F S C c ) . ...................11 3 . 2 F ir st in v ari a nt s o f cy c l e s . ..................... 14 4 J oi n t i n v aria nt s : o r t hogo na l i ty a nd inv e r s ions 16 4 . 1 In v a r ian t o r t hog o n ali t y ty p e c o n di t ion s . ...................16 4.2 Inversions in cycles . ..................... 1 9 4 . 3 F oc al o rt hog on a lit y . ..................... 2 1 5 M et ri c pro per t ies f r o m cy c l e inva ria n t s 24 5 . 1 Dista n c es a n d len gt hs . ..................... 24 Sym metry comma I n tegra b i l i ty a nd Geometry : .... 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