Timeline for Second Betti number of lattices in $\mathrm{SL}_3(\mathbf{R})$
Current License: CC BY-SA 4.0
7 events
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| Nov 5, 2018 at 17:26 | comment | added | YCor | Actually, I have checked that the only finite order, non-identity automorphisms of $\mathrm{SL}_1(D)$ have order 3, and for each such automorphism, the connected component of the group of fixed points is a 2-dimensional $\mathbf{Q}$-anisotropic torus— so the arithmetic subgroups have $\chi=0$. (The group of algebraic automorphisms of $\mathrm{SL}_1(D)$, for $D$ central division algebra of degree 3, is reduced to $\mathrm{PGL}_1(D)$, i.e., is not twice bigger.) | |
| Oct 29, 2018 at 20:27 | vote | accept | YCor | ||
| Oct 29, 2018 at 17:45 | comment | added | Steffen Kionke | Yes, you need a finite order automorphism of $\mathrm{SL}_1(D)$ defined over $\mathbb{Q}$ such that the group of fixed points $G$ has positive dimension and its arithmetic subgroups have non-zero Euler characteristic. (As a remark: all automorphisms of $D$ are inner by Skolem-Noether, but I don't see how one can write down useful elements in $D$.) | |
| Oct 29, 2018 at 17:14 | comment | added | YCor | Thanks. If I understand correctly, the cocompact lattices commensurability-conjugation correspond to $\mathbf{Q}$-anisotropic $\mathbf{R}$-split $\mathbf{Q}$-forms of $\mathrm{SL}_3$. These come into two families: $\mathrm{SU}(h,L/F)$ as you consider above, and the other family are the $\mathrm{SL}_1(D)$ where $D$ is a central division algebra of degree 3, itself $\mathbf{Q}$-anisotropic and $\mathbf{R}$-split. So one has to find a non-identity finite order $\mathbf{Q}$-defined automorphism of $\mathrm{SL}_1(D)$? (Having positive-dimensional set of fixed points is not an issue.) | |
| Oct 29, 2018 at 16:24 | comment | added | Steffen Kionke | @YCor: The example on p.721 refers only to the split algebraic group $\mathrm{SL}_3$ defined over $\mathbb{Q}$, i.e. to certain non-cocompact lattices. Cocompact lattices are defined using other algebraic groups (e.g. $\mathrm{SU}(h,L/F)$ as above) and one needs to find a suitable automorphism for each of these groups. The exponent $3/8$ is the ratio of the dimension of the fixed point group $\mathrm{SO}(2,1)$ and the dimension of $\mathrm{SL}_3(\mathbb{R})$. In the non-cocompact cases above one can do slightly better (and find a $4$-dim fixed point group). | |
| Oct 29, 2018 at 16:03 | comment | added | YCor | Great, thank you. Your $\Gamma_n$ are, in addition, torsion-free by construction. A side question: in your article, beginning of page 13 (published, p721) you write "So every arithmetic subgroup of $\mathrm{SL}_3$ contains a cofinal $p$-tower (for every odd prime p) with some Betti number growing faster than the square root of the index." Here you write the exponent $3/8$ (and not $1/2$) and you say that you "did not check the other examples of cocompact lattices in $\mathrm{SL}_3(\mathbf{R})$". How can I conciliate these statements? | |
| Oct 29, 2018 at 15:05 | history | answered | Steffen Kionke | CC BY-SA 4.0 |