arXiv subjects

Tatiana Abdelnaim

Publications and source records attributed to Tatiana Abdelnaim.

3 recordsLinked to original sources

arXiv2026

Double Steinberg coinvariants for special linear groups

For a field $F$ we study the coinvariants for the $SL_n(F)$-action on the double Steinberg module $St_n(F) \otimes St_n(F)$ and show they have a rich algebraic structure: for $n = 2$ it is the Grothendieck-Witt group of $F$, and for all $n$ they assemble to a graded nonunital $\mathbb{Z}[F^\times]$-algebra, whose rational (underived) indecomposables may be expressed in terms of the augmentation ideal of the Grothendieck-Witt group. We then explain, building on work of Galatius-Kupers-Randal-Williams, that the special linear groups $SL_n(F)$ assemble to an $E_\infty$-algebra in a suitable functor category, whose $E_2$-homology groups have a vanishing line of slope 2 and on the critical line are given by the double Steinberg coinvariants.

math.AT↗

arXiv2026

Codimensions one and two cohomology of Hecke congruence subgroups

For $n\geq 1$ and $p$ a prime, the Hecke congruence subgroup $Γ_{0,n}(p)\leq \mathrm{SL}_n(\mathbb{Z})$ is the subgroup of matrices whose first column is of the form $(*,0,\dots,0)^t\bmod p$. Borel--Serre showed that $Γ_{0,n}(p)$ has virtual cohomological dimension $\binom{n}{2}$. The first author proved that the rational cohomology in this top degree $\binom{n}{2}$ vanishes for $n$ sufficiently large compared to $p$. We prove analogous results in codimension $1$ and $2$.

math.AT↗

arXiv2026

Top-dimensional rational cohomology of the congruence subgroup $Γ_{0,n}^+(p)$

Let $Γ_{0,n}^+(p)\subset \mathrm{SL}_n(\mathbb{Z})$ be the congruence subgroup of level-$p$ whose first column is of the form $(*,0,\dots,0)^t\bmod p$. We prove that the top-dimensional cohomology group $H^{\binom{n}{2}}(Γ_{0,n}^+(p);\mathbb{Q})$ vanishes for $p\in\{2,3,5,7,13\}$ if $n \geq 3$, as well as for $p \leq 6n-14$. Additionally, we prove a non-vanishing result, showing that this cohomology group is nonzero for $n = 2$ for every prime $p$, and for $n=3$ for all primes $p \notin \{2,3,5,7,13\}$.

math.AT↗