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Pablo Sánchez-Peralta

Publications and source records attributed to Pablo Sánchez-Peralta.

8 recordsLinked to original sources
arXiv2026

Simon's knot genus problem and Lewin $3$-manifold groups

We provide a positive answer to an old problem of Jonathan K. Simon: if $K$ and $K'$ are two knots such that there is an epimorphism from the knot group of $K$ to the knot group of $K'$, then the genus of $K$ is greater than or equal to the genus of $K'$. This result follows from our proof of a conjecture of Friedl and Lück. Specifically, we show that any map between admissible $3$-manifolds that induces an epimorphism on the fundamental groups and an isomorphism on the rational homologies yields an inequality of Thurston norms. Under these conditions, we also establish new upper bounds on the number of faces of the Thurston norm ball of the target manifold. To resolve Friedl and Lück's conjecture, we prove that locally indicable $3$-manifold groups are Lewin groups, thereby confirming a conjecture of Jaikin-Zapirain within the class of $3$-manifold groups. As a further consequence of our methods, we show that the crossed product of a division ring and a torsion-free $3$-manifold group that is virtually free-by-cyclic is a pseudo-Sylvester domain.

math.GT↗
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

Coherent RFRS groups

We prove that a finitely generated virtually RFRS group of cohomological dimension at most $2$ is coherent if and only if its second $L^{2}$-Betti number vanishes if and only if it is virtually free-by-cyclic. The non-vanishing of the second $L^{2}$-Betti number provides the first known global obstruction to coherence in any reasonably wide class of groups, allowing for proofs of incoherence without needing to exhibit explicit witnesses to incoherence. As applications of this result, we completely characterise coherence among two-dimensional Coxeter groups, confirming conjectures of Jankiewicz and Wise, and show that incoherence is generic in groups of nonpositive deficiency, confirming a conjecture of Wise. We also find that, among virtually compact special groups of virtual cohomological dimension two, coherence is algorithmically decidable and is a quasi-isometry, measure equivalence, and profinite invariant. In an appendix, Marco Linton applies one of the main results to prove that cubulated locally quasi-convex hyperbolic groups are virtually free-by-cyclic, solving problems of Abdenbi--Wise and Wise in the cubulated case.

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

Universal localizations, Atiyah conjectures and graphs of groups

Let $G$ be a countable group that is the fundamental group of a graph of groups with finite edge groups and vertex groups that satisfy the strong Atiyah conjecture over $K \subseteq \mathbb{C}$ a field closed under complex conjugation. Assume that the orders of finite subgroups of $G$ are bounded above. We show that $G$ satisfies the strong Atiyah conjecture over $K$. In particular, this implies that the strong Atiyah conjecture is closed under free products. Moreover, we prove that the $\ast$-regular closure of $K[G]$ in $\mathcal{U}(G)$, $\mathcal{R}_{\small K[G]}$, is a universal localization of the graph of rings associated to the graph of groups, where the rings are the corresponding $\ast$-regular closures. As a result, we obtain that the algebraic and center-valued Atiyah conjecture over $K$ are also closed under the graph of groups construction provided that the edge groups are finite. We also infer some consequences on the structure of the $K_0$ and $K_1$-groups of $\mathcal{R}_{\small K[G]}$. The techniques developed allow us to prove that $K[G]$ fulfills the strong, algebraic and center-valued Atiyah conjectures and that $\mathcal{R}_{\small K[G]}$ is the universal localization of $K[G]$ over the set of all matrices that become invertible in $\mathcal{U}(G)$ if $G$ lies in a certain class of groups $\mathcal{T}_{\small \mathcal{VLI}}$, which contains in particular virtually-{locally indicable} groups that are the fundamental group of a graph of virtually free groups.

math.GR↗
arXiv2025

Division rings for group algebras of virtually compact special groups and $3$-manifold groups

Let $k$ be a division ring and let $G$ be either a torsion-free virtually compact special group or a finitely generated torsion-free $3$-manifold group. We embed the group algebra $kG$ in a division ring and prove that the embedding is Hughes-free whenever $G$ is locally indicable. In particular, we prove that Kaplansky's Zero Divisor Conjecture holds for all group algebras of torsion-free $3$-manifold groups. The embedding is also used to confirm a conjecture of Kielak and Linton. Thanks to the work of Jaikin-Zapirain and Linton, another consequence of the embedding is that $kG$ is coherent whenever $G$ is a virtually compact special one-relator group. If $G$ is a torsion-free one-relator group, let $\overline{kG}$ be the division ring containing $kG$ constructed by Lewin and Lewin. We prove that $\overline{kG}$ is Hughes-free whenever a Hughes-free $kG$-division ring exists. This is always the case when $k$ is of characteristic zero; in positive characteristic, our previous result implies that this happens when $G$ is virtually compact special.

math.GR↗
arXiv2024

On vanishing criteria of $L^2$-Betti numbers of groups

Let $G$ be a countable group and $k$ a positive integer, we show that the $L^2$-Betti numbers of the group $G$ vanish up to degree $k$ provided that there is some infinite index subgroup $H$ with finite $k$th $L^2$-Betti number containing a normal subgroup of $G$ whose $L^2$-Betti numbers are all zero below degree $k$. This generalizes previous criteria of both Sauer and Thom, and Peterson and Thom. In addition, we exhibit a purely algebraic proof of a well-known theorem of Gaboriau concerning the first $L^2$-Betti number which was requested by Bourdon, Martin and Valette. Finally, we provide evidence of a positive answer for a question posted by Hillman that wonders whether the above statement holds for $k = 1$ and $H$ containing a subnormal subgroup instead.

math.GR↗