arXiv subjects
Publications and source records attributed to Martin Nitsche.
Existing property (T) proofs for $Aut(F_n)$, $n\geq 4$, rely crucially on extensive computer calculations. We give a new proof that $Aut(F_n)$ has property (T) for all but finitely many $n$ that is inspired by the semidefinite programming approach but does not use the computer in any step. More specifically, we prove property (T) for a certain extension $\Gamma_n$ of $SAut(F_n)$ as $n\to\infty$.
Based on a result of Singh--Venkataramana, Bajpai--Dona--Singh--Singh gave a criterion for a discrete Zariski-dense subgroup of Sp(2n,Z) to be a lattice. We adapt this criterion so that it can be used in some situations that were previously excluded. We apply the adapted method to subgroups of Sp(6,Z) and Sp(4,Z) that arise as the monodromy groups of hypergeometric differential equations. In particular, we show that out of the 40 maximally unipotent Sp(6) hypergeometric groups more than half are arithmetic, answering a question of Katz in the negative.
This article studies the orthogonal hypergeometric groups of degree five. We establish the thinness of 12 out of the 19 hypergeometric groups of type O(3,2) from [4, Table 6]. Some of these examples are associated with Calabi-Yau 4-folds. We also establish the thinness of 9 out of the 17 hypergeometric groups of type O(4,1) from [12], where the thinness of 7 other cases was already proven. The O(4,1) type groups were predicted to be all thin and our result leaves just one case open.
We explore the thinness of hypergeometric groups of type $\mathrm{Sp}(4)$ and $\mathrm{Sp}(6)$ by applying a new approach of computer-assisted ping pong. We prove the thinness of $17$ hypergeometric groups with maximally unipotent monodromy in $\mathrm{Sp}(6)$, completing the classification of all $40$ such groups into arithmetic and thin cases. In addition, we establish the thinness of further $46$ hypergeometric groups in $\mathrm{Sp}(6)$, and of $3$ hypergeometric groups in $\mathrm{Sp}(4)$, completing the classification of all $\mathrm{Sp}(4)$ hypergeometric groups. To the best of our knowledge, this article produces the first $63$ examples in the cyclotomic family of Zariski dense non-arithmetic hypergeometric monodromy groups of real rank three.
For $M$ a compact Riemannian manifold Brandenbursky and Marcinkowski constructed a transfer map $H_b^*(\pi_1(M))\to H_b^*(Homeo_{vol,0}(M))$ and used it to show that for certain $M$ the space $\overline{EH}_b^3(Homeo_{vol,0}(M))$ is infinite-dimensional. Kimura adapted the argument to $Diff_{vol}(D^2,\partial D^2)$. We extend both results to the higher degrees $\overline{EH}_b^{2n}$, $n\geq 1$. We also show that for certain $M$ the ordinary cohomology $H^*(Homeo_{vol,0}(M))$ is non-trivial in all degrees. In our computations we view the transfer map as being induced by a coupling of groups.
We show that the semidefinite programs involved in the computer proofs for Kazhdan's property (T) satisfy strong duality and that the dual programs have a geometric interpretation in terms of harmonic cocycles. By dualizing geometric arguments about cocycles, we are able to simplify the property (T) SDP in the case where it carries a symmetry by finite-order inner automorphisms. As an application, we simplify the SDP proof for $SL(n,\mathbb{Z})$ and we prove that $Aut(F_4)$ has property (T).
Let $N \subset M$ be a submanifold embedding of spin manifolds of some codimension $k \geq 1$. A classical result of Gromov and Lawson, refined by Hanke, Pape and Schick, states that $M$ does not admit a metric of positive scalar curvature if $k = 2$ and the Dirac operator of $N$ has non-trivial index, provided that suitable conditions are satisfied. In the cases $k=1$ and $k=2$, Zeidler and Kubota, respectively, established more systematic results: There exists a transfer $\mathrm{KO}_\ast(\mathrm{C}^{\ast} \pi_1 M)\to \mathrm{KO}_{\ast - k}(\mathrm{C}^\ast \pi_1 N)$ which maps the index class of $M$ to the index class of $N$. The main goal of this article is to construct analogous transfer maps $E_\ast(\mathrm{B}\pi_1M) \to E_{\ast-k}(\mathrm{B}\pi_1N)$ for different generalized homology theories $E$ and suitable submanifold embeddings. The design criterion is that it is compatible with the transfer $E_\ast(M) \to E_{\ast-k}(N)$ induced by the inclusion $N \subset M$ for a chosen orientation on the normal bundle. Under varying restrictions on homotopy groups and the normal bundle, we construct transfers in the following cases in particular: In ordinary homology, it works for all codimensions. This slightly generalizes a result of Engel and simplifies his proof. In complex K-homology, we achieve it for $k \leq 3$. For $k \leq 2$, we have a transfer on the equivariant KO-homology of the classifying space for proper actions.
We give a new criterion for solvability of group equations, providing proofs of various generalizations of the Kervaire-Laudenbach conjecture for Connes-embeddable groups.