Endomorphisms of Artin groups of typeD

We determine a classification of the endomorphisms of the Artin group AOED n of type D n for n 6.In particular we determine its automorphism group and its outer automorphism group.We also determine a classification of the homomorphisms from AOED n to the Artin group AOEA n 1 of type A n 1 and a classification of the homomorphisms from AOEA n 1 to AOED n for n 6.We show that any endomorphism of the quotient AOED n =Z.AOED n / lifts to an endomorphism of AOED n for n 4. We deduce a classification of the endomorphisms of AOED n =Z.AOED n /, we determine the automorphism and outer automorphism groups of AOED n =Z.AOED n /, and we show that AOED n =Z.AOED n / is co-Hopfian for n 6.The results are algebraic in nature but the proofs are based on topological arguments (curves on surfaces and mapping class groups).