14
$\begingroup$

Let $G$ be a group and $R$ be a commutative ring. Recall that a derivation of the group ring $R[G]$ is a map $\delta : R[G] \rightarrow R[G]$ such that

$$\delta(x+y)=\delta(x)+\delta(y) \quad \text{and} \quad \delta(xy) = \delta(x) \epsilon(y) + x \delta(y)$$

for $x,y \in R[G]$. Here $\epsilon : R[G] \rightarrow R$ is the augmentation map.

There are a lot of derivations of $R[G]$ if $G$ is a free group, as here we have the Fox free differential calculus.

Question : Do there exist interesting derivations on the group ring (analogous in some way to the Fox free derivatives) for groups that are not free? I'm especially interested in fundamental groups of closed surfaces.

Improve this question
$\endgroup$
3
  • 1
    $\begingroup$ (Outer) derivations are the same as the first Hochschild cohomology group, and there is lot of works devoted to Hochschild (co)homology of group rings. How interesting it is, is in the eyes of beholder, but by any rate most of these works are, probably, far from the spirit of Fox's differential calculus. $\endgroup$
    – Pasha Zusmanovich
    Commented Aug 8, 2011 at 14:35
  • $\begingroup$ The question was completely answered in in Andreas Thom's answer and John_L did not visit MO for close to 2 years. I think, the question can be closed now as "no longer relevant". $\endgroup$
    – Misha
    Commented Jun 14, 2013 at 14:10
  • 1
    $\begingroup$ I rolled back the edit. The "differential calculus" tag does not seem appropriate here. I also (along with Misha) voted to close as "no longer relevant". $\endgroup$
    – Andy Putman
    Commented Jun 14, 2013 at 16:17

1 Answer 1

Reset to default
17
$\begingroup$

There are always derivations of the form $\delta_b(a) = ab - b\varepsilon(a)$ for $b \in R[G]$, which are called inner derivations. Derivations modulo inner derivations are classified by the first group cohomology with coefficients in the $G$-module $R[G]$, $H^1(G,R[G])$.

In the special case $R=\mathbb Z_2$, Stallings defined a group to have infinitely many ends if $H^1(G,\mathbb Z_2[G])$ has dimension $\geq 2$, in which case it is automatically infinite dimensional. A group has two ends $H^1(G,\mathbb Z_2[G])$ is one-dimensional and one end if $H^1(G,\mathbb Z_2[G])$ vanishes. (See John Stallings, On Torsion-Free Groups with Infinitely Many Ends, Ann. of Math., 1968 vol. 88 (2) pp. 312-334).

Stallings proved that a finitely generated group $G$ has more than one end if and only if the group $G$ admits a nontrivial decomposition as an amalgamated free product or an HNN extension over a finite subgroup. (See Wikipedia.) In that sense, there is a perfect understanding of the class of groups you are looking for.

Fundamental groups of surfaces have one end, so that there exist no interesting derivations, at least for $R=\mathbb Z_2$. Computations for PID's are similar, see Section 13.5 in

Ross Geoghegan, Topological Methods in Group Theory (Graduate Texts in Mathematics), Springer.

This book also contains a whole chapter about the structure of $H^*(G,R[G])$ and its relation to the the topology of the universal cover $EG$ of the classifying space of $G$.

Improve this answer
$\endgroup$

Not the answer you're looking for? Browse other questions tagged .