ahalfozhs alakkunna anumaanam

ganithashaasthrathil, ippol oru sidhaanthamaaya aalfozhs anumaanam, parimithamayi srishtikkappetta clainian grouppinte paridhi koottam onnukil muzhuvan riman golamo allengil alavu 0 aanennu prasthaavikkunnu. 
clainian groupinu parimithamaaya ennam vashangalulla oru adisthaana domaine undennu theliyikkunnayaalaan eee anumaanam avatharippichathu. topologically teyim cheytha clainian grouppukal jyamitheeyamaayi teyim cheythathaanennu kaanichukondu topologically teyim cheytha clainian grouppukalkkaayulla aalfozhs anumaanam theliyichu, athinaal parimithamayi srishtikkappetta adisthaana grouppukalulla hyperbolic 3-maanifoldukal topologically teyim cheythirikkunnu (compact 3-maanifoldukalude inteeriyarinu homomorphic) enna maardante tainus anumaanathil ninnaan aalfozhs anumaanam pinthudarunnathu. eee randaamathe anumaanam swathanthramaayi, arcum arcum theliyikkapettu.
 paridhi settu muzhuvan golamaakumbol, paridhi settile clainian grouppinte pravartthanam ergodic aanennum kaanichu.

