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Weibel's "Algebraic K-theory of rings of integers in local and global fields" says $K_{4k}(\mathbb{Z})$ are known to have odd order, with no prime factors less than $10^7$, but are conjectured to be zero.

Since the paper is from 2004 I was wondering if anything new is known about those groups.

I'm interested on any new research on them.

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    $\begingroup$ According to the schedule for "Groups in Galway 2014", there was a talk by Philippe Elbaz-Vincent entitled "The group $K_8(\Bbb Z)$ is trivial". I was unable to find any further information. maths.nuigalway.ie/conferences/gig14 $\endgroup$
    – Tyler Lawson
    Commented Nov 16, 2014 at 5:51
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    $\begingroup$ As a follow-up on Tyler Lawson's comment - as far as I understand, the strategy for computation of K-theory of $\mathbb{Z}$ is to compute group homology of $GL_N(\mathbb{Z})$ (via Voronoi cell decomposition of the symmetric space), as outlined in arxiv.org/abs/math/0207067. Some more details on the computations for $N=5,6,7$ are found in arxiv.org/abs/1001.0789. I would expect that the same techniques, pushed further, are behind the announced triviality of $K_8(\mathbb{Z})$. $\endgroup$
    – Matthias Wendt
    Commented Nov 16, 2014 at 10:58
  • $\begingroup$ Published and somewhat extended version of arxiv.org/abs/math/0207067 is sciencedirect.com/science/article/pii/S0001870813002223 where it is stated at the end that "The group $K_8(\mathbb{Z})$ still remains unknown". $\endgroup$
    – Zurab Silagadze
    Commented Nov 24, 2014 at 4:40

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$K_8(\mathbb{Z})=0$

A proof of the result announced by Elbaz-Vincent and mentioned above can now be found in:

  • Mathieu Dutour-Sikirić, Philippe Elbaz-Vincent, Alexander Kupers, and Jacques Martinet, Voronoi complexes in higher dimensions, cohomology of $GL_N(\mathbb{Z})$ for $N\geq8$ and the triviality of $K_8(\mathbb{Z})$, arXiv:1910.11598v1 (submitted on October 25, 2019)

A shorter proof, which (unlike the one above) does not require additional computer calculations, is presented in:

The very last remark in Kupers’s short note addresses the case of $K_{12}(\mathbb{Z})$:

Remark 2.6. If Conjecture 2 of [CFP14] were true with coefficients in $\mathbb{Z}[1/((n+1)!)]$ instead of $\mathbb{Q}$, it could be used to prove that $K_{12}(\mathbb{Z})=0$ in a similar manner.

[CFP14] Thomas Church, Benson Farb, and Andrew Putman, A stability conjecture for the unstable cohomology of $SL_n(\mathbb{Z})$, mapping class groups, and $Aut(F_n)$, Algebraic topology: applications and new directions, 55–70, Contemp. Math., vol. 620, Amer. Math. Soc., 2014, MR3290086

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Well, the consensus seems to be that this is an open problem, for $k >1$.

This is a quote from A. Raghuram's paper on the volume "The Bloch–Kato Conjecture for the Riemann Zeta Function" (page 8), published in april 2015.

[...] it is expected $K_{4a}(\mathbb{Z})=0$. This is proven for $a=1$ and is open as yet for $a\geq 2$.

Given that other contributors include Stephen Lichtenbaum and Manfred Kolster, this most likely represents the current knowledge at least at the time of the conference (2012).

If someone has any update (particularly regarding Philippe Elbaz-Vincent's claim, see the comments above), feel free to answer or comment.

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As you probably know, this is related to the Vandiver conjecture. So the references at http://en.wikipedia.org/wiki/Kummer%E2%80%93Vandiver_conjecture might help.

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