The lower algebraic $K$-theory of virtually cyclic subgroups of the braid groups of the sphere and of $\mathbb{Z}[B_4(\mathbb{S}^2)]$

John Guaschi (1, 2, 3, 4) , Daniel Juan-Pineda (5) , Silvia Millán-López (6)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme
2 UNICAEN - Université de Caen Normandie
3 NU - Normandie Université
4 CNRS - Centre National de la Recherche Scientifique
5 CCM - Centro de Ciencias Matemáticas UNAM Morelia
6 Colegio de Bachilleres del Estado de Tlaxcala

We study $K$-theoretical aspects of the braid groups $B_n(\mathbb{S}^{2})$ on $n$ strings of the $2$-sphere, which by results of the second two authors, are known to satisfy the Farrell-Jones fibred isomorphism conjecture~\cite{JM}. In light of this, in order to determine the algebraic $K$-theory of the group ring $\mathbb{Z}[B_n(\mathbb{S}^{2})]$, one should first compute that of its virtually cyclic subgroups, which were classified by D.~L.~Gonçalves and the first author. We calculate the Whitehead and $K_{-1}$-groups of the group rings of the finite subgroups (dicyclic and binary polyhedral) of $B_n(\mathbb{S}^{2})$ for all $4\leq n\leq 11$. Some new phenomena occur, such as the appearance of torsion for the $K_{-1}$-groups. We then go on to study the case $n=4$ in detail, which is the smallest value of $n$ for which $B_n(\mathbb{S}^{2})$ is infinite. We show that $B_n(\mathbb{S}^{2})$ is an amalgamated product of two finite groups, from which we are able to determine a universal space for proper actions of the group $B_n(\mathbb{S}^{2})$. We also calculate the algebraic $K$-theory of the infinite virtually cyclic subgroups of $B_n(\mathbb{S}^{2})$, including the Nil groups of the quaternion group of order $8$. This enables us to determine the lower algebraic $K$-theory of $\mathbb{Z}[B_n(\mathbb{S}^{2})]$.

Mots clés: Braid groups; Virtually cyclic subgroups; algebraic K-theory; Nil groups

Domaines: K-théorie et homologie [math.KT]; Théorie des groupes [math.GR]; Topologie géométrique [math.GT]

John Guaschi, Daniel Juan-Pineda, Silvia Millán-López. The lower algebraic $K$-theory of virtually cyclic subgroups of the braid groups of the sphere and of $\mathbb{Z}[B_4(\mathbb{S}^2)]$. Springer International Publishing, 2018, SpringerBriefs in Mathematics, 978-3-319-99488-8. ⟨10.1007/978-3-319-99489-5⟩. ⟨hal-00734244v2⟩