arXiv subjects

Andrei Jaikin-Zapirain

Publications and source records attributed to Andrei Jaikin-Zapirain.

At least 19 recordsLinked to original sources

arXiv2026

Thurston norm, polytopes and splitting complexity

Let $G$ be a finitely generated torsion-free group satisfying the Strong Atiyah Conjecture. Assuming that $b_1^{(2)}(G)=0$, we associate to each surjective integral character $ϕ$ the quantity $b_1^{(2)}(\ker ϕ)$. We prove that the resulting function extends to a seminorm on $H^1(G;\mathbb{R})$ and is the thickness function of a polytope in $H_1(G;\mathbb{R})$. More generally, assuming that $G$ is of type $\mathrm{FP}_k(\mathbb{Q})$ and has $b_k^{(2)}(G)=0$, we prove that the function assigning $b_k^{(2)}(\ker ϕ)$ to each surjective integral character $ϕ$ is likewise the thickness function of a polytope. This confirms a conjecture of Friedl, Lück, and Tillmann. For torsion-free virtually (finitely generated free)-by-cyclic groups, we relate the resulting invariant to the $L^2$-Betti complexity of a character, thereby confirming a conjecture of Gardam and Kielak. As an application, we establish the existence of an algorithm for computing the Bieri--Neumann--Strebel invariant of free-by-cyclic groups and discuss connections with the isomorphism problem for such groups.

math.GR↗

arXiv2026

Average Numbers of Homomorphisms to Random Modules over Free Group Algebras

Let $F$ be a finitely generated free group, and let $L$ be a finitely presented $\mathbb{F}_q[F]$-module. We study the average number $Λ_L(n)$ of $\mathbb{F}_q[F]$-module homomorphisms from $L$ to an $\mathbb{F}_q[F]$-module of dimension $n$ over $\mathbb{F}_q$. We show that, for all sufficiently large $n$, the quantity $Λ_L(n)$ is given by a rational function of $q^n$ and satisfies \[ Λ_L(n) = q^{χ(L)n} + \sum_{N\in A(L)} q^{χ(L/N)n}\bigl(1+O(q^{-n})\bigr), \] where $χ(L)$ denotes the Euler characteristic of $L$, and $A(L)$ is the set of nonzero $\mathbb{F}_q[F]$-submodules $N$ of $L$ that have no nonzero free quotients. Our proof is based on a theory of partial modules that may be of independent interest and have further applications.

math.GR↗

arXiv2026

Profinite detection of free products and free factors

Let $G$ be the fundamental group of a graph of finitely generated virtually free groups with virtually cyclic edge groups. We shaw that $G$ is cohomologically good if $G$ is residually finite. If $G$ is LERF, we prove that G splits non-trivially as a free product if and only if its profinite completion $\widehat{G}$ splits non-trivially as a free profinite product. Moreover, we are able to detect one-ended free factors of $G$ from $\widehat{G}$. As an application, we deduce that any profinitely rigid word in a finitely generated free group is universally profinitely rigid.

math.GR↗

arXiv2026

Sylvester domains and pro-$p$ groups

Let $G$ be a finitely generated torsion-free pro-$p$ group containing an open free-by-$\mathbb{Z}_p$ pro-$p$ subgroup. We show that the completed group algebra of $G$ over $\mathbb{F}_p$ is a Sylvester domain. Moreover the inner rank of a matrix $A$ over this completed group algebra can be calculated by approximation by ranks corresponding to finite quotients of $G$, that is, if $G=G_1>G_2>\ldots$ is a chain of normal open subgroups of $G$ with trivial intersection and $A_i$ is the matrix over $\mathbb{F}_p[G/G_i]$ obtained from the matrix $A$ by applying the natural homomorphism induced from $G \to G/G_i$, then the inner rank of $A$ equals $\lim_{i\to \infty} \frac{\operatorname{rk}_{\mathbb{F}_p} (A_i)}{|G:G_i|}$. As a consequence, we obtain a particular case of the mod $p$ Lück approximation for abstract finitely generated subgroups of free-by-$\mathbb{Z}_p$ pro-$p$ groups.

math.GR↗

arXiv2026

Free $\mathbb Q$-groups are residually torsion-free nilpotent

We develop a method to show that some (abstract) groups can be embedded into a free pro-$p$ group. In particular, we show that a finitely generated subgroup of a free $\mathbb Q$-group can be embedded into a free pro-$p$ group for almost all primes $p$. This solves an old problem raised by G. Baumslag: free $\mathbb Q$-groups are residually torsion-free nilpotent.

math.GR↗

arXiv2026

Some remarks on Grothendieck pairs

We revisit the paper of Alexander Grothendiek where he introduced Grothendieck pairs and discuss the relation between profinite rigidity and left/right Grothendieck rigidity. We also show that various groups are left and/or right Grothendieck rigid and, in particular, all ascending HNN extensiona of finitely generated free groups are right Grothendieck rigid. Along the way we present a number of questions and suggestions for further research.

math.GR↗

arXiv2026

Free groups are $L^2$-subgroup rigid

In this paper, we introduce the notion of $L^2$-subgroup rigid groups and demonstrate that free groups are $L^2$-subgroup rigid. As a consequence, we establish the equivalence between compressibility, inertness, strong inertness, and $L^2$-independence for a finitely generated subgroup of a free group, confirming a conjecture by Dicks and Ventura as well as the one by Antolin and Jaikin-Zapirain.

math.GR↗

arXiv2025

Group pairs, coherence and Farrell--Jones Conjecture for $K_0$

A group pair $(G, X)$ consists of a group $G$ together with a $G$-set $X$. Such a pair encodes properties of $G$ relative to the stabilisers of points in $X$. In this paper, we show how to combine properties of group pairs and their stabilisers to prove coherence results for $G$ and its group algebra, as well as to study the quotient of $G$ obtained by killing the stabilisers. In particular, we prove that a torsion-free one-relator product of locally indicable groups is coherent provided that both factor groups are coherent. Moreover, we show that the group algebra of such a group over a field of characteristic $0$ is coherent whenever the group algebras of the factors are coherent. As other consequences of our methods, we also show that extensions of coherent locally indicable hyperbolic groups by $\mathbb{Z}$ are coherent and that groups admitting a Cohen--Lyndon presentation satisfy the Farrell--Jones Conjecture for $K_{0}$.

math.GR↗

arXiv2025

On the coherence of one-relator groups and their group algebras

We prove that one-relator groups are coherent, solving a well-known problem of Gilbert Baumslag. Our proof strategy is readily applicable to many classes of groups of cohomological dimension two. We show that fundamental groups of two-complexes with non-positive immersions are homologically coherent, we show that groups with staggered presentations and many Coxeter groups are coherent and we show that group algebras over fields of characteristic zero of groups with reducible presentations without proper powers are coherent.

math.GR↗

arXiv2024

Asymptotic invariants of residually finite just infinite groups

Recently, Eduard Schesler and the second author constructed examples of finitely generated residually finite, hereditarily just infinite groups with positive first $L^2$-Betti number. In contrast to their result, we show that a finitely generated residually-$p$ just infinite group has trivial first $L^2$-Betti number. Moreover, we prove that the normal homology rank gradient of a finitely generated, residually finite, just infinite group vanishes.

math.GR↗

arXiv2024

Twisted $L^2$-Betti numbers of sofic groups

Wolfang Lück asked if twisted $L^2$-Betti numbers of a group are equal to the usual $L^2$-Betti numbers rescaled by the dimension of the twisting representation. We confirm this for sofic groups.

math.GR↗

arXiv2024

Parafree graphs of groups with cyclic edge groups

We establish a combination theorem for parafree groups. These groups were introduced by Baumslag in the sixties. One of the current motivations for a better understanding of their structure is that they show up naturally in connection with Remeslennikov's conjecture on the profinite rigidity of free groups. In this article, we determine when the fundamental group of a finite graph of groups with cyclic edge groups is parafree.

math.GR↗

arXiv2024

Prosolvable rigidity of surface groups

Surface groups are known to be the Poincaré Duality groups of dimension two since the work of Eckmann, Linnell and Müller. We prove a prosolvable analogue of this result that allows us to show that surface groups are profinitely (and prosolvably) rigid among finitely generated groups that satisfy $\mathrm{cd}(G)=2$ and $b_2^{(2)}(G)=0$. We explore two other consequences. On the one hand, we derive that if $u$ is a surface word of a finitely generated free group $F$ and $v\in F$ is measure equivalent to $u$ in all finite solvable quotients of $F$ then $u$ and $v$ belong to the same $\mathrm{Aut}(F)$-orbit. Finally, we get a partial result towards Mel'nikov's surface group conjecture. Let $F$ be a free group of rank $n\geq 3$ and let $w\in F$. Suppose that $G=F/\langle\!\langle w\rangle\!\rangle$ is a residually finite group all of whose finite-index subgroups are one-relator groups. Then $G$ is 2-free. Moreover, we show that if $H^2(G; \mathbb{Z})\neq 0$ then $G$ must be a surface group.

math.GR↗

arXiv2022

Free factors and profinite completions

Can one detect free products of groups via their profinite completions? We answer positively among virtually free groups. More precisely, we prove that a subgroup of a finitely generated virtually free group $G$ is a free factor if and only if its closure in the profinite completion of $G$ is a profinite free factor. This generalises results by Parzanchevski and Puder (later also proved by Wilton) for free groups. Our methods are entirely different to theirs, combining homological properties of profinite groups and the decomposition theory of Dicks and Dunwoody.

math.GR↗

arXiv2020

On the space of Sylvester matrix rank functions

Given a ring $R$, the notion of Sylvester rank function was conceived within the context of Cohn's classification theory of epic division $R$-rings. In this paper we study and describe the space of Sylvester rank functions on certain families of rings, including Dedekind domains, simple left noetherian rings and skew Laurent polynomial rings $\mathcal{D}[t^{\pm 1};τ]$ for any division ring $\mathcal{D}$ and any automorphism $τ$ of $\mathcal{D}$.

math.RA↗

arXiv2019

The strong Atiyah and Lück approximation conjectures for one-relator groups

It is shown that the strong Atiyah conjecture and the Lück approximation conjecture in the space of marked groups hold for locally indicable groups. In particular, this implies that one-relator groups satisfy both conjectures. We also show that the center conjecture, the independence conjecture and the strong eigenvalue conjecture hold for these groups. As a byproduct we prove that the group algebra of a locally indicable group over a field of characteristic zero has a Hughes-free epic division algebra and, in particular, it is embedded in a division algebra.

math.GR↗

arXiv2019

The Hanna Neumann conjecture for Demushkin Groups

We confirm the Hanna Neumann conjecture for topologically finitely generated closed subgroups $U$ and $W$ of a nonsolvable Demushkin group $G$. Namely, we show that \begin{equation*} \sum_{g \in U \backslash G/W} \bar d(U \cap gWg^{-1}) \leq \bar d(U) \bar d(W) \end{equation*} where $\bar d(K) = \max\{d(K) - 1, 0\}$ and $d(K)$ is the least cardinality of a topological generating set for the group $K$.

math.GR↗

arXiv2016

Finite 2-groups with odd number of conjugacy classes

In this paper we consider finite 2-groups with odd number of real conjugacy classes. On one hand we show that if $k$ is an odd natural number less than 24, then there are only finitely many finite 2-groups with exactly $k$ real conjugacy classes. On the other hand we construct infinitely many finite 2-groups with exactly 25 real conjugacy classes. Both resuls are proven using pro-$p$ techniques and, in particular, we use the Kneser classification of semi-simple $p$-adic algebraic groups.

math.GR↗