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Claudio Llosa Isenrich

Publications and source records attributed to Claudio Llosa Isenrich.

At least 19 recordsLinked to original sources
arXiv2026

On the Dehn functions of central products of nilpotent groups

We determine the Dehn functions of central products of two families of filiform nilpotent Lie groups of arbitrary dimension with all simply connected nilpotent Lie groups with cyclic centre and strictly lower nilpotency class. We also determine the Dehn functions of all central products of nilpotent Lie groups of dimension at most $5$ with one-dimensional centre. This confirms a conjecture of Llosa Isenrich, Pallier and Tessera for these cases, providing further evidence that the Dehn functions of central products are often strictly lower than those of the factors. Our work generalises previous results of Llosa Isenrich, Pallier and Tessera and produces an uncountable family of nilpotent Lie groups without lattices whose Dehn functions are strictly lower than the ones of the associated Carnot-graded groups. A consequence of our main result is the existence of an infinite family of groups such that Cornulier's bounds on the $e$ for which there is an $O(r^e)$-bilipschitz equivalence between them and their Carnot-graded groups are asymptotically optimal, as the nilpotency class goes to infinity.

math.GR↗
arXiv2026

A Brown Theorem for Dehn functions of graphs of groups

We prove an upper bound on the Dehn function of a group $G$ acting cellularly, cocompactly, and without inversions on a simply connected CW complex $X$ in terms of the Dehn functions of the vertex stabilizers, the Dehn function of $X$, and the distortion of the edge stabilizers, provided that $X$ is either a tree or the stabilizer of each $2$-cell has finite index in the stabilizer of every edge in its boundary. This provides a Dehn function analogue of Brown's Theorem for finiteness properties and an answer to a question of Zaremsky in these cases. We also prove analogues of our result for higher Dehn functions when $X$ is a tree.

math.GR↗
arXiv2025

Infinitely presented simple groups separated by homological finiteness properties

Given a finitely generated linear group $G$ over $\mathbb{Q}$, we construct a simple group $Γ$ that has the same finiteness properties as $G$ and admits $G$ as a quasi-retract. As an application, we construct a simple group of type $\mathrm{FP}_{\infty}$ that is not finitely presented. Moreover we show that for every $n \in \mathbb{N}$ there is a simple group of type $\mathrm{FP}_n$ that is neither finitely presented nor of type $\mathrm{FP}_{n+1}$. Since our simple groups arise as Röver--Nekrashevych groups, this answers a question of Zaremsky.

math.GR↗
arXiv2025

$L^p$ measure equivalence of nilpotent groups

We classify compactly generated locally compact groups of polynomial growth up to $L^p$ measure equivalence (ME) for all $p\leq 1$. To achieve this, we combine rigidity results (previously proved for discrete groups by Bowen and Austin) with new constructions of explicit orbit equivalences between simply connected nilpotent Lie groups. In particular, we prove that for every pair of simply connected nilpotent Lie groups there is an $L^p$ orbit equivalence for some $p>0$, where we can choose $p>1$ if and only if the groups have isomorphic asymptotic cones. We also prove analogous results for lattices in simply connected nilpotent Lie groups. This yields a strong converse of Austin's Theorem that two nilpotent groups which are $L^1$ ME have isomorphic Carnot graded groups. We also address the much harder problem of extending this classification to $L^p$ ME for $p>1$: we obtain the first rigidity results, providing examples of nilpotent groups with isomorphic Carnot graded groups (hence $L^1$ OE) which are not $L^p$ ME for some finite (explicit) $p$. For this we introduce a new technique, which consists of combining induction of cohomology and scaling limits via the use of a theorem of Cantrell. Finally, in the appendix, we extend theorems of Bowen, Austin and Cantrell on $L^1$ ME to locally compact groups.

math.GR↗
arXiv2025

Dehn functions of subgroups of products of free groups. Part II: Precise computations

We prove that the Bridson-Dison group has quartic Dehn function, thereby providing the first precise computation of the Dehn function of a subgroup of a direct product of free groups with super-quadratic Dehn function. We also prove that coabelian subgroups of direct products of $n$ free groups of finiteness type $\mathcal{F}_{n-1}$ and of corank $r\leq n-2$ have quadratic Dehn functions.

math.GR↗
arXiv2025

Dehn functions of subgroups of products of free groups. Part I: Uniform upper bounds

Subgroups of direct products of finitely many finitely generated free groups form a natural class that plays an important role in geometric group theory. Its members include fundamental examples, such as the Stallings-Bieri groups. This raises the problem of understanding their geometric invariants. We prove that finitely presented subgroups of direct products of three free groups, as well as subgroups of finiteness type $\mathcal{F}_{n-1}$ in a direct product of $n$ free groups, have Dehn function bounded above by $N^9$. This gives a positive answer to a question of Dison within these important subclasses and provides new insights in the context of Bridson's conjecture stating that finitely presented subgroups of direct products of free groups have polynomially bounded Dehn function. To prove our results we generalise techniques for "pushing fillings" into normal subgroups.

math.GR↗
arXiv2025

On the Boone--Higman Conjecture for groups acting on locally finite trees

We develop a method for proving the Boone--Higman Conjecture for groups acting on locally finite trees. As a consequence, we prove the Boone--Higman Conjecture for all Baumslag--Solitar groups and for all free(finite rank)-by-cyclic groups, solving it in two cases that have been raised explicitly by Belk, Bleak, Matucci and Zaremsky. We also illustrate that our method has applications beyond these cases and may offer a route for proving the Boone--Higman Conjecture for many classes of groups.

math.GR↗
arXiv2025

Profinite rigidity of Kähler groups: Riemann surfaces and subdirect products

This paper establishes strong profinite rigidity results for Kähler groups, showing that certain groups are determined within the class of residually finite Kähler groups by their profinite completion. Examples include products of surface groups and certain groups with exotic finiteness properties studied earlier by Dimca-Papadima-Suciu and Llosa Isenrich. Consequently, there are aspherical smooth projective varieties that are determined up to homeomorphism by their algebraic fundamental group. The main tool is the following: the holomorphic fibrations of a closed Kähler manifold over hyperbolic 2-orbifolds can be recovered from the profinite completion of its fundamental group. We also prove profinite invariance of the BNS invariant.

math.GT↗
arXiv2025

Explicit polynomial bounds on Dehn functions of subgroups of hyperbolic groups

In 1999 Brady constructed the first example of a non-hyperbolic finitely presented subgroup of a hyperbolic group by fibring a non-positively curved cube complex over the circle. We show that his example has Dehn function bounded above by $n^{96}$. This provides the first explicit polynomial upper bound on the Dehn function of a finitely presented non-hyperbolic subgroup of a hyperbolic group. We also determine the precise hyperbolicity constant for the $1$-skeleton of the universal cover of the cube complex in Brady's construction with respect to the $4$-point condition for hyperbolicity.

math.GR↗
arXiv2024

Subgroups of CAT(0) groups, exotic finiteness properties and non-QI-embeddings into linear groups

For every positive integer $n$ we construct an example of a subgroup $L< G$ of a linear ${\rm CAT}(0)$ group $G$ such that $L$ is of finiteness type $\mathcal{F}_{n-1}$ and not $\mathcal{F}_n$, and $L$ does not admit a representation into $\mathsf{GL}_d(k)$ which is a quasi-isometric embedding for any local field $k$. We further prove that there is a faithful representation of $L$ into some $\mathsf{GL}_{\ell}(\mathbb{C})$ which is not the restriction of any representation of $G$. This generalises a family of fibre products of type $\mathcal{F}_2$ not $\mathcal{F}_3$ with these properties constructed by the second author.

math.GR↗
arXiv2024

From the second BNSR invariant to Dehn functions of coabelian subgroups

Given a finitely presented group $G$ and a surjective homomorphism $G\to \mathbb{Z}^n$ with finitely presented kernel $K$, we give an upper bound on the Dehn function of $K$ in terms of an area-radius pair for $G$. As a consequence we obtain that finitely presented coabelian subgroups of hyperbolic groups have polynomially bounded Dehn function. This generalises results of Gersten and Short and our proof can be viewed as a quantified version of results from Renz' thesis on the second BNSR invariant.

math.GR↗
arXiv2023

Groups with exotic finiteness properties from complex Morse theory

Recent constructions have shown that interesting behaviours can be observed in the finiteness properties of Kähler groups and their subgroups. In this work, we push this further and exhibit, for each integer $k$, new hyperbolic groups admiting surjective homomorphisms to $\mathbb{Z}$ and to $\mathbb{Z}^{2}$, whose kernel is of type $\mathscr{F}_{k}$ but not of type $\mathscr{F}_{k+1}$. By a fibre product construction, we also find examples of nonnormal subgroups of Kähler groups with exotic finiteness properties.

math.GT↗
arXiv2023

Cone-equivalent nilpotent groups with different Dehn functions

For every $k\geqslant 3$, we exhibit a simply connected $k$-nilpotent Lie group $N_k$ whose Dehn function behaves like $n^k$, while the Dehn function of its associated Carnot graded group $\mathsf{gr}(N_k)$ behaves like $n^{k+1}$. This property and its consequences allow us to reveal three new phenomena. First, since those groups have uniform lattices, this provides the first examples of pairs of finitely presented groups with bilipschitz asymptotic cones but with different Dehn functions. The second surprising feature of these groups is that for every even integer $k \geqslant 4$ the centralized Dehn function of $N_k$ behaves like $n^{k-1}$ and so has a different exponent than the Dehn function. This answers a question of Young. Finally, we turn our attention to sublinear bilipschitz equivalences (SBE). Introduced by Cornulier, these are maps between metric spaces inducing bi-Lipschitz homeomorphisms between their asymptotic cones. These can be seen as weakenings of quasiisometries where the additive error is replaced by a sublinearly growing function $v$. We show that a $v$-SBE between $N_k$ and $\mathsf{gr}(N_k)$ must satisfy $v(n)\succcurlyeq n^{1/(2k + 2)}$, strengthening the fact that those two groups are not quasiisometric. This is the first instance where an explicit lower bound is provided for a pair of SBE groups.

math.GR↗
arXiv2023

Dehn functions of coabelian subgroups of direct products of groups

We develop new methods for computing the precise Dehn functions of coabelian subgroups of direct products of groups, that is, subgroups which arise as kernels of homomorphisms from the direct product onto a free abelian group. These improve and generalise previous results by Carter and Forester on Dehn functions of level sets in products of simply connected cube complexes, by Bridson on Dehn functions of cocyclic groups and by Dison on Dehn functions of coabelian groups. We then provide several applications of our methods to subgroups of direct products of free groups, to groups with interesting geometric finiteness properties and to subgroups of direct products of right-angled Artin groups.

math.GR↗
arXiv2022

Subgroups of hyperbolic groups, finiteness properties and complex hyperbolic lattices

We prove that in a cocompact complex hyperbolic arithmetic lattice $Γ< {\rm PU}(m,1)$ of the simplest type, deep enough finite index subgroups admit plenty of homomorphisms to $\mathbb{Z}$ with kernel of type $\mathscr{F}_{m-1}$ but not of type $\mathscr{F}_{m}$. This provides many finitely presented non-hyperbolic subgroups of hyperbolic groups and answers an old question of Brady. Our method also yields a proof of a special case of Singer's conjecture for aspherical Kähler manifolds.

math.GR↗
arXiv2021

Complex hypersurfaces in direct products of Riemann surfaces

We study smooth complex hypersurfaces in direct products of closed hyperbolic Riemann surfaces and give a classification in terms of their fundamental groups. This answers a question of Delzant and Gromov on subvarieties of products of Riemann surfaces in the smooth codimension one case. We also answer Delzant and Gromov's question of which subgroups of a direct product of surface groups are Kähler for two classes: subgroups of direct products of three surface groups; and subgroups arising as kernel of a homomorphism from the product of surface groups to $\mathbb{Z}^3$. These results will be a consequence of answering the more general question of which subgroups of a direct product of surface groups are the image of a homomorphism, which is induced by a holomorphic map, for the same two classes. This provides new constraints on Kähler groups.

math.GT↗