arXiv subjects

Jonathan Fruchter

Publications and source records attributed to Jonathan Fruchter.

6 recordsLinked to original sources

arXiv2025

Virtual homological torsion: abundance versus growth in books of $I$-bundles

Let $\mathcal{B}$ be a book of $I$-bundles, all of whose pages are surfaces of negative Euler characteristic. In this short note, we prove that torsion in the first homology of $\mathcal{B}$ grows subexponentially in the index along any exhausting tower of regular finite-sheeted covers. By contrast, recent work of Ascari and the author shows that, apart from the obvious exceptions, $\mathcal{B}$ has abundant virtual homological torsion, which can grow exponentially along exhausting towers of non-regular finite covers.

math.GT↗

arXiv2025

Homological torsion growth in non-normal chains of graphs of free groups

Let $G$ be a hyperbolic group that splits as a graph of free groups with cyclic edge groups, and which is not isomorphic to a free product of free and surface groups. We show that $G$ admits an exhausting, nested sequence of finite-index non-normal subgroups $G\ge G_1 \ge G_2 \ge \cdots$ with exponential homological torsion growth. More specifically, we prove that simultaneously for every prime $p$, $\liminf_{n\rightarrow \infty} \frac{\log \vert \mathrm{Tor}_p(G_n^{\mathrm{ab}})\vert}{[G:G_n]} >0$ (where $\mathrm{Tor}_p(G_n^{\mathrm{ab}}) = \{g \in G_n^{\mathrm{ab}} \;\vert\; g \text{ has order a power of } p\}$).

math.GR↗

arXiv2025

Virtual homological torsion in graphs of free groups with cyclic edge groups

Let $G$ be a hyperbolic group that splits as a graph of free groups with cyclic edge groups. We prove that, unless $G$ is isomorphic to a free product of free and surface groups, every finite abelian group $M$ appears as a direct summand in the abelianization of some finite-index subgroup $G'\le G$. As an application, we deduce that free products of free and surface groups are profinitely rigid among hyperbolic graphs of free groups with cyclic edge groups. We also conclude that partial surface words in a free group are determined by the word measures they induce on finite groups.

math.GR↗

arXiv2022

Virtual homology of limit groups and profinite rigidity of direct products

We show that the virtual second Betti number of a finitely generated, residually free group $G$ is finite if and only if $G$ is either free, free abelian or the fundamental group of a closed surface. We also prove a similar statement in higher dimensions. We then develop techniques involving rank gradients of pro-$p$ groups, which allow us to recognise direct product decompositions. Combining these ideas, we show that direct products of free and surface groups are profinitely rigid among finitely presented, residually free groups, partially resolving a conjecture of Bridson's. Other corollaries that we obtain include a confirmation of Mel'nikov's surface group conjecture in the residually free case, and a description of closed aspherical manifolds of dimension at least $5$ with a residually free fundamental group.

math.GR↗

arXiv2021

Limit groups over coherent right-angled Artin groups are cyclic subgroup separable

We prove that cyclic subgroup separability is preserved under exponential completion for groups that belong to a class that includes all coherent RAAGs and toral relatively hyperbolic groups; we do so by exploiting the structure of these completions as iterated free products with commuting subgroups. From this we deduce that the cyclic subgroups of limit groups over coherent RAAGs are separable, answering a question of Casals-Ruiz, Duncan and Kazachov. We also discuss relations between free products with commuting subgroups and the word problem, and recover the fact that limit groups over coherent RAAGs and toral relatively hyperbolic groups have a solvable word problem.

math.GR↗

arXiv2020

Formal solutions and the first-order theory of acylindrically hyperbolic groups

We generalise Merzlyakov's theorem about the first-order theory of non-abelian free groups to all acylindrically hyperbolic groups. As a corollary, we deduce that if $G$ is an acylindrically hyperbolic group and $E(G)$ denotes the unique maximal finite normal subgroup of $G$, then $G$ and the HNN extension $G\dot{\ast}_{E(G)}$, which is simply the free product $G\ast\mathbb{Z}$ when $E(G)$ is trivial, have the same $\forall\exists$-theory. As a consequence, we prove the following conjecture, formulated by Casals-Ruiz, Garreta and de la Nuez Gonz\'alez: acylindrically hyperbolic groups have trivial positive theory. In particular, one recovers a result proved by Bestvina, Bromberg and Fujiwara, stating that, with only the obvious exceptions, verbal subgroups of acylindrically hyperbolic groups have infinite width.

math.GR↗