arXiv subjects

Nadav Gropper

Publications and source records attributed to Nadav Gropper.

3 recordsLinked to original sources

arXiv2026

A Neukirch-Uchida Theorem for 3-Manifolds

The classical Neukirch-Uchida theorem states that the absolute Galois group determines a number field up to isomorphism. We prove an analogue of this theorem for 3-manifolds in the framework of arithmetic topology. We study infinite links in 3-manifolds that behave like the set of primes, satisfying a Chebotarev density property. Relative to such a stably Chebotarev link, we define the absolute Galois group of a 3-manifold as the inverse limit of profinite completions of finite sublink complements. Our main result shows that two branched covers of the three-sphere over a stably Chebotarev link are homeomorphic if and only if their absolute Galois groups are isomorphic via a characteristic-preserving isomorphism. The proof translates the key ideas from the number-theoretic argument into topology, relying on Hilbert ramification theory for infinite covers and local-global principles. In doing so, it also provides a systematic justification for viewing Chebotarev links as the precise topological analogue of prime numbers in anabelian geometry. In addition, we discuss further conditions for links to play the role of prime numbers

math.GT↗

arXiv2025

Arithmetic field theory via pro-p duality groups

Using the theory of pro-p groups and relative Poincar\'{e} duality, we define a type of cobordism category well suited to arithmetic topology. We completely classify topological quantum field theories on these two-dimensional versions of our cobordism categories. This classification uses Frobenius algebras with extra operations corresponding to automorphisms of the p-adic integers. We look in more detail at the example of arithmetic Dijkgraff--Witten theory for a finite gauge p-group in this setting. This allows us to deduce formulae counting Galois extensions of local p-adic fields whose Galois groups are the given gauge group.

math.NT↗

arXiv2023

Surfaces and p-adic fields I: Dehn twists

Following the philosophy of arithmetic topology, we describe a point of view which helps look at surfaces and $p$-adic fields in a "uniform way", and show that results on mapping class groups can be extended to this point of view, and thus be applied to $G_{K}$, the absolute Galois groups of the $p$-adic field $K$. By moving both groups to the world of pro-$p$ groups (for $G_{K}$ we take its maximal pro-$p$ quotient, and for $\pi_{1}(S)$ we take its pro-$p$ completion), we see they both are pro-$p$ Poincare duality groups of dimension $2$, also known as Demuskin groups. Such groups have a very nice classification in terms of generators and relations. By using the language of graphs of groups and examining discrete groups with Demuskin type relations, we show that all splittings of a Demuskin group come from a discrete splitting, which in turn helps us show that Dehn twists make sense in such a context. This gives us a family of infinite order Outer automorphisms of $G_{K}(p)$ the maximal pro-$p$ quotient of the Galois group, which are "arithmetic Dehn twists". On the other hand, when specializing this to Demuskin groups coming from surface groups, one gets back the usual definition of Dehn twists on surfaces. As a finally corollary, we show that there in an infinite family of non-isomorphic discrete groups, having isomorphic pro-$l$ completions for all primes $l$ (which are free pro-$l$ for $l \neq p$ and Demuskin for $l=p$).

math.NT↗