PhD Defense Tim Korte Berland
Title: On Analytic Torsion in Families of Arithmetic Manifolds
Abstract:
This thesis concerns the asymptotics of analytic torsion when varying over families of arithmetic manifolds, and applies this to study the growth of torsion in the cohomology of the manifolds. The thesis is split into three chapters, the first being a preliminary section, the second an article on this topic for the group $G = \operatorname{SL}(n)$, and the third presenting ongoing work on generalising the results of the article to a larger family of reductive groups.
The main results of Chapter $2$ is the following: Let $\Gamma(N)$ be the principal congruence subgroup in $\operatorname{SL}(n, \mathbb{Z})$ of level $N$, and let $X(N)$ be the associated locally symmetric space. Let $\tau$ be a finite-dimensional irreducible representation of $ \operatorname{SL}(n, \mathbb{R})$. Assume that $\tau$ is $\lambda$-strongly acyclic for a certain $\lambda >0$. Then, as $N$ goes to infinity, the analytic torsion $T_{X(N)}(\tau)$ of $X(N)$ is equal to the $L^2$-torsion $T^{(2)}_{X(N)}(\tau)$ of $X(N)$ up to an error term of the size $O(vol(X(N))N^{−(n−1)}(log N)^a)$ for some $a > 0$. We furthermore prove the existence of infinitely many $\lambda$-strongly acyclic representations of any semisimple Lie group of deficiency greater than or equal to $1$.
In chapter $3$, this theorem is appropriately generalised to principal congruence subgroups in reductive groups. This result is then applied to estimate the growth of torsion in the cohomology of the Borel-Serre compactification of congruence quotients of manifolds of the form $(\mathbb{H}_2)^b \times (\mathbb{H}_3)^c$.
To attend the digital defence, please follow the link:
https://ucph-ku.zoom.us/j/61958665442?pwd=QRit2e44sPQbPmaNmEdwRgFw8aaC4l.1
Principal supervisor:
Associate Professor, Jasmin Matz
Email: matz@math.ku.dk
Assessment committee
(Chairperson) Professor Morten Risager
Department of Mathematical Sciences, SCIENCE
Email: risager@math.ku.dk
Professor Haluk Sengun
University of Sheffield
Email: m.sengun@sheffield.ac.uk
Professor (emeritus) Werner Mueller
University of Bonn
Email: mueller@math.uni-bonn.de