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Let be the level- principal congruence subgroup of . Borel and Serre proved that the cohomology of vanishes above degree . We study the cohomology in this top degree . Let denote the Tits building of . Lee and Szczarba conjectured that is isomorphic to and proved that this holds for . We partially prove and partially disprove this conjecture by showing that a natural map is always surjective, but is only injective for . In particular, we completely calculate and improve known lower bounds for the ranks of for .
Miller, Jeremy 1 ; Patzt, Peter 1 ; Putman, Andrew 2
@article{GT_2021_25_2_a8, author = {Miller, Jeremy and Patzt, Peter and Putman, Andrew}, title = {On the top-dimensional cohomology groups of congruence subgroups of {SL(n,} {\ensuremath{\mathbb{Z}})}}, journal = {Geometry & topology}, pages = {999--1058}, publisher = {mathdoc}, volume = {25}, number = {2}, year = {2021}, doi = {10.2140/gt.2021.25.999}, url = {http://geodesic.mathdoc.fr/articles/10.2140/gt.2021.25.999/} }
TY - JOUR AU - Miller, Jeremy AU - Patzt, Peter AU - Putman, Andrew TI - On the top-dimensional cohomology groups of congruence subgroups of SL(n, ℤ) JO - Geometry & topology PY - 2021 SP - 999 EP - 1058 VL - 25 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/gt.2021.25.999/ DO - 10.2140/gt.2021.25.999 ID - GT_2021_25_2_a8 ER -
%0 Journal Article %A Miller, Jeremy %A Patzt, Peter %A Putman, Andrew %T On the top-dimensional cohomology groups of congruence subgroups of SL(n, ℤ) %J Geometry & topology %D 2021 %P 999-1058 %V 25 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/gt.2021.25.999/ %R 10.2140/gt.2021.25.999 %F GT_2021_25_2_a8
Miller, Jeremy; Patzt, Peter; Putman, Andrew. On the top-dimensional cohomology groups of congruence subgroups of SL(n, ℤ). Geometry & topology, Tome 25 (2021) no. 2, pp. 999-1058. doi : 10.2140/gt.2021.25.999. http://geodesic.mathdoc.fr/articles/10.2140/gt.2021.25.999/
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