Let $p$ be an odd prime number and $G$ a finitely generated pro-$p$ group. Define $I(G)$ the augmentation ideal of the group algebra of $G$ over $F_p$ and define the Hilbert series of $G$ by: $G(t):=sum_{n\in \NN} \dim_{\F_p} I^n(G)/I^{n+1}(G)$.The series $G(t)$ gives several information on $G$. First, during the $60$'s, Golod and Shafarevich used Hilbert series to relate the number of generators and relations defining $G$, to the cardinality of $G$. Also, if $G(t)$ satisfies some equalities, we can read the cohomological dimension of $G$. Between 1980 and 2000's, Anick and Labute, introduced a sufficient and easy condition on the relations of pro-$p$-group $G$, such that $G(t)$ satisfies some equality ensuring that $G$ is of cohomological dimension less than two. Groups satisfying this sufficient condition are called mild.If we consider $K$ a number field, then using mild groups, we can construct extensions$L/K$, in which infinitely many primes split completely, and with prescribed Hilbert series Gal(L/K)(t). Consequently Gal(L/K) is finitely generated, infinitely presented and of cohomological dimension $2$.
De Mehmet Haluk Sengun
De Mehmet Haluk Sengun
De Aurel Page
De Alexander Hulpke
De Alexander Hulpke
De Alexander Hulpke
De Paul Gunnells
De Aurel Page
De Peter Patzt
De Peter Patzt
De Peter Patzt
De Jennifer Wilson
De Jennifer Wilson
De Jennifer Wilson
De Haowen Zhang
De Renaud Coulangeon
De Renaud Coulangeon
De Bill Allombert
De Lewis Combes
De Alexander Hulpke
De Alexander Hulpke
De Paul Gunnells
De Paul Gunnells
De Paul Gunnells
De Paul Gunnells
De Graham Ellis
De Graham Ellis
De Graham Ellis
De Graham Ellis
De Graham Ellis
De Philippe Elbaz-Vincent
De Philippe Elbaz-Vincent
De Philippe Elbaz-Vincent
De Bettina Eick
De Petru Constantinescu
De Angelica Babei
De Alexander, D. Rahm
De Benjamin Brück
De Calista Bernard
De Philippe Elbaz-Vincent
De Bettina Eick
De Bettina Eick
De Bettina Eick
De Bettina Eick
De Herbert Gangl
De Zachary Himes
