On a conjecture of E. Rapaport Strasser about knot-like groups and its pro-p version |
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Authors: | Dessislava H Kochloukova |
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Institution: | UNICAMP-IMECC, Cx. P. 6065, 13083-970 Campinas, SP, Brazil |
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Abstract: | A group G is knot-like if it is finitely presented of deficiency 1 and has abelianization G/G′?Z. We prove the conjecture of E. Rapaport Strasser that if a knot-like group G has a finitely generated commutator subgroup G′ then G′ should be free in the special case when the commutator G′ is residually finite. It is a corollary of a much more general result : if G is a discrete group of geometric dimension n with a finite K(G,1)-complex Y of dimension n, Y has Euler characteristics 0, N is a normal residually finite subgroup of G, N is of homological type FPn-1 and G/N?Z then N is of homological type FPn and hence G/N has finite virtual cohomological dimension vcd(G/N)=cd(G)-cd(N). In particular either N has finite index in G or cd(N)?cd(G)-1.Furthermore we show a pro-p version of the above result with the weaker assumption that G/N is a pro-p group of finite rank. Consequently a pro-p version of Rapaport's conjecture holds. |
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Keywords: | 20J05 |
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