# A lower bound on the top degree rational cohomology of the symplectic group of a number ring

Zachary Himes

Let R be a number ring. If one fixes i and lets n go to infinity, then the rational cohomology H^i(SL_n(R); Q) stabilizes in a range.

Outside this range, little is known about the rational cohomology in general except that it vanishes for all i > \nu_n, where \nu_n is an explicit constant described by Borel--Serre. For i=\nu_n, Church--Farb--Putman recently showed that the dimension of H^{\nu_n}(SL_n(R); Q)$ is at least (|Cl(R)| -1)^{n-1}, where Cl(R) denotes the class group of R. For the rational cohomology of the symplectic group Sp_{2n}(R), similar stability and vanishing patterns occur. In joint work with Benjamin Br\"uck, we obtain a similar lower bound for the the top degree rational cohomology of Sp_{2n}(R) and show it has dimension at least (|Cl(R)| -1)^n.

Outside this range, little is known about the rational cohomology in general except that it vanishes for all i > \nu_n, where \nu_n is an explicit constant described by Borel--Serre. For i=\nu_n, Church--Farb--Putman recently showed that the dimension of H^{\nu_n}(SL_n(R); Q)$ is at least (|Cl(R)| -1)^{n-1}, where Cl(R) denotes the class group of R. For the rational cohomology of the symplectic group Sp_{2n}(R), similar stability and vanishing patterns occur. In joint work with Benjamin Br\"uck, we obtain a similar lower bound for the the top degree rational cohomology of Sp_{2n}(R) and show it has dimension at least (|Cl(R)| -1)^n.

Building: | East Hall |
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Event Type: | Presentation |

Tags: | Mathematics |

Source: | Happening @ Michigan from Representation Stability Seminar - Department of Mathematics, Department of Mathematics |