22 results on '"Isenrich, Claudio Llosa"'
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2. Dehn functions of subgroups of products of free groups: 3-factor case, $F_{n-1}$ case, and Bridson-Dison group
- Author
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Ascari, Dario, Bertolotti, Federica, Italiano, Giovanni, Isenrich, Claudio Llosa, and Migliorini, Matteo
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Mathematics - Group Theory ,Mathematics - Geometric Topology ,20F65 (Primary) 20F05, 20F67, 20F69, 57M07 (Secondary) - Abstract
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 that finitely presented subgroups of direct products of free groups have polynomially bounded Dehn function. We also give the first precise computation of a superquadratic Dehn function of a finitely presented subgroup of a direct product of finitely many free groups: we show that the Bridson-Dison group is a subgroup of a direct product of three free groups with Dehn function $N^4$. To prove our results we generalise techniques for "pushing fillings" into normal subgroups and define a new invariant for obtaining optimal lower bounds on Dehn functions., Comment: 77 pages, 4 figures
- Published
- 2024
3. From the second BNSR invariant to Dehn functions of coabelian subgroups
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Isenrich, Claudio Llosa
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Mathematics - Group Theory ,Mathematics - Geometric Topology ,20F65, 20F67, 20F69, 20F06, 20J05, 20F05, 57M07 - Abstract
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., Comment: 14 pages, 5 figures
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- 2024
4. On the Dehn functions of central products of nilpotent groups
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García-Mejía, Jerónimo, Isenrich, Claudio Llosa, and Pallier, Gabriel
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Mathematics - Group Theory ,Mathematics - Geometric Topology ,Mathematics - Metric Geometry ,20F69, 20F18 - Abstract
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., Comment: 56 pages, 4 figures, 2 tables. Comments welcome. v2: new Theorem VI and moved Appendix B.2 into a new Section 6
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- 2023
5. Groups with exotic finiteness properties from complex Morse theory
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Isenrich, Claudio Llosa and Py, Pierre
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Mathematics - Geometric Topology ,Mathematics - Complex Variables ,Mathematics - Differential Geometry ,Mathematics - Group Theory - Abstract
Recent constructions have shown that interesting behaviours can be observed in the finiteness properties of K\"ahler 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 K\"ahler groups with exotic finiteness properties., Comment: 23 pages
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- 2023
6. Finitely presented kernels of homomorphisms from hyperbolic groups onto free abelian groups
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Kropholler, Robert and Isenrich, Claudio Llosa
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Mathematics - Group Theory ,20F67, 20F65, 20J05, 20F05, 57M07 - Abstract
For every $m\geq 2$ we produce an example of a non-hyperbolic finitely presented subgroup $H < G$ of a hyperbolic group $G$, which is the kernel of a surjective homomorphism $\phi: G\to \mathbb{Z}^m$. The examples we produce are of finiteness type $F_2$ and not $F_3$., Comment: 15 pages
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- 2023
7. Subgroups of hyperbolic groups, finiteness properties and complex hyperbolic lattices
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Isenrich, Claudio Llosa and Py, Pierre
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Mathematics - Group Theory ,Mathematics - Complex Variables ,Mathematics - Differential Geometry ,Mathematics - Geometric Topology ,20F67 (Primary) 20F65, 20J05, 32J27, 57M07 (Secondary) - Abstract
We prove that in a cocompact complex hyperbolic arithmetic lattice $\Gamma < {\rm PU}(m,1)$ 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 K\"ahler manifolds., Comment: 22 pages
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- 2022
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8. Hyperbolic groups containing subgroups of type $\mathscr{F}_{3}$ not $\mathscr{F}_{4}$
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Isenrich, Claudio Llosa, Martelli, Bruno, and Py, Pierre
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Mathematics - Group Theory ,Mathematics - Geometric Topology ,20F67 (Primary), 20F65, 57M07, 20J05, 20F05 (Secondary) - Abstract
We give examples of hyperbolic groups which contain subgroups that are of type $\mathscr{F}_{3}$ but not of type $\mathscr{F}_{4}$. These groups are obtained by Dehn filling starting from a non-uniform lattice in ${\rm PO}(8,1)$ which was previously studied by Italiano, Martelli and Migliorini., Comment: 24 pages, 1 figure
- Published
- 2021
9. Dehn functions of coabelian subgroups of direct products of groups
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Kropholler, Robert and Isenrich, Claudio Llosa
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Mathematics - Group Theory ,20F65 (20F05, 20F06, 20F69) - Abstract
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., Comment: 25 pages, 6 figures, v3: Minor corrections and improvements to the exposition. Final version
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- 2021
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10. Cone-equivalent nilpotent groups with different Dehn functions
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Isenrich, Claudio Llosa, Pallier, Gabriel, and Tessera, Romain
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Mathematics - Group Theory ,Mathematics - Differential Geometry ,Mathematics - Geometric Topology ,Mathematics - Metric Geometry ,20F69, 20F18 (Primary), 20F65, 20F05, 51F30, 22E25, 57T10 (Secondary) - Abstract
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., Comment: 64 pages.v3: final version, minor corrections and improvements to the exposition
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- 2020
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11. Mapping class groups, multiple Kodaira fibrations, and CAT(0) spaces
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Isenrich, Claudio Llosa and Py, Pierre
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Mathematics - Geometric Topology ,Mathematics - Algebraic Geometry ,Mathematics - Complex Variables ,Mathematics - Group Theory - Abstract
We study several geometric and group theoretical problems related to Kodaira fibrations, to more general families of Riemann surfaces, and to surface-by-surface groups. First we provide constraints on Kodaira fibrations that fiber in more than two distinct ways, addressing a question by Catanese and Salter about their existence. Then we show that if the fundamental group of a surface bundle over a surface is a ${\rm CAT}(0)$ group, the bundle must have injective monodromy (unless the monodromy has finite image). Finally, given a family of closed Riemann surfaces (of genus $\ge 2$) with injective monodromy $E\to B$ over a manifold $B$, we explain how to build a new family of Riemann surfaces with injective monodromy whose base is a finite cover of the total space $E$ and whose fibers have higher genus. We apply our construction to prove that the mapping class group of a once punctured surface virtually admits injective and irreducible morphisms into the mapping class group of a closed surface of higher genus., Comment: 32 pages, v3. The order of the sections has changed. This is the final version, to be published by Math. Annalen
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- 2020
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12. Residually free groups do not admit a uniform polynomial isoperimetric function
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Isenrich, Claudio Llosa and Tessera, Romain
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Mathematics - Group Theory ,20F65 (Primary), 20F05, 20F69 (Secondary) - Abstract
We show that there is no uniform polynomial isoperimetric function for finitely presented subgroups of direct products of free groups, by producing a sequence of subgroups $G_r\leq F_2^{(1)} \times \dots \times F_2^{(r)}$ of direct products of 2-generated free groups with Dehn functions bounded below by $n^{r}$. The groups $G_r$ are obtained from the examples of non-coabelian subdirect products of free groups constructed by Bridson, Howie, Miller and Short. As a consequence we obtain that residually free groups do not admit a uniform polynomial isoperimetric function., Comment: 9 pages, v2: Improvements to the exposition and minor corrections. Final accepted version, to appear in the Proceedings of the American Mathematical Society
- Published
- 2018
13. On the Dehn functions of K\'ahler groups
- Author
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Isenrich, Claudio Llosa and Tessera, Romain
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Mathematics - Geometric Topology ,Mathematics - Algebraic Geometry ,Mathematics - Group Theory ,32J27, 20F65 - Abstract
We address the problem of which functions can arise as Dehn functions of K\"ahler groups. We explain why there are examples of K\"ahler groups with linear, quadratic, and exponential Dehn function. We then proceed to show that there is an example of a K\"ahler group which has Dehn function bounded below by a cubic function and above by $n^6$. As a consequence we obtain that for a compact K\"ahler manifold having non-positive holomorphic bisectional curvature does not imply having quadratic Dehn function., Comment: 15 pages, V2: Minor corrections and improvements to the exposition. Final accepted version, to appear in Groups, Geometry, and Dynamics
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- 2018
14. Complex hypersurfaces in direct products of Riemann surfaces
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Isenrich, Claudio Llosa
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Mathematics - Geometric Topology ,Mathematics - Algebraic Geometry ,Mathematics - Group Theory - Abstract
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 K\"ahler 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 K\"ahler groups., Comment: 14 pages, V3: Some changes to the exposition, particularly in the introduction. Also added some new references
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- 2018
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15. K\'ahler groups and subdirect products of surface groups
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Isenrich, Claudio Llosa
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Mathematics - Geometric Topology ,Mathematics - Algebraic Geometry ,Mathematics - Group Theory ,32J27, 20F65 (Primary), 32Q15, 20J05 (Secondary) - Abstract
We present a construction that produces infinite classes of K\"ahler groups that arise as fundamental groups of fibres of maps to higher dimensional tori. Following the work of Delzant and Gromov, there is great interest in knowing which subgroups of direct products of surface groups are K\"ahler. We apply our construction to obtain new classes of irreducible, coabelian K\"ahler subgroups of direct products of $r$ surface groups. These cover the full range of possible finiteness properties of irreducible subgroups of direct products of $r$ surface groups: For any $r\geq 3$ and $2\leq k \leq r-1$, our classes of subgroups contain K\"ahler groups that have a classifying space with finite $k$-skeleton while not having a classifying space with finitely many $(k+1)$-cells. We also address the converse question of finding constraints on K\"ahler subdirect products of surface groups and, more generally, on homomorphisms from K\"ahler groups to direct products of surface groups. We show that if a K\"ahler subdirect product of $r$ surface groups admits a classifying space with finite $k$-skeleton for $k>\frac{r}{2}$, then it is virtually the kernel of an epimorphism from a direct product of surface groups onto a free abelian group of even rank., Comment: 30 pages, V4: Some results were strengthened (in particular, Theorems 1.2 and 1.5 now include some additional consequences). Minor corrections and improvements to the exposition. Final accepted version, to appear in Geometry & Topology
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- 2017
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16. Kodaira fibrations, K\'ahler groups, and finiteness properties
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Bridson, Martin R. and Isenrich, Claudio Llosa
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Mathematics - Geometric Topology ,Mathematics - Algebraic Geometry ,Mathematics - Group Theory ,32J27, 20J05 (Primary), 32Q15, 20F65 (Secondary) - Abstract
We construct classes of K\"ahler groups that do not have finite classifying spaces and are not commensurable to subdirect products of surface groups. Each of these groups is the fundamental group of the generic fibre of a holomorphic map from a product of Kodaira fibrations onto an elliptic curve., Comment: 21 pages, 1 figure, V2: Minor corrections and improvements to the exposition. Final accepted version, to appear in the Transactions of the American Mathematical Society
- Published
- 2016
17. Kähler groups and Geometric Group Theory
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Isenrich, Claudio Llosa and Bridson, Martin R.
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516.3 ,Mathematics ,Geometry and Topology ,Geometric Group Theory ,Ka¨hler groups ,Complex geometry - Abstract
In this thesis we study Kähler groups and their connections to Geometric Group Theory. This work presents substantial progress on three central questions in the field: (1) Which subgroups of direct products of surface groups are Kähler? (2) Which Kähler groups admit a classifying space with finite (n-1)-skeleton but no classifying space with finitely many n-cells? (3) Is it possible to give explicit finite presentations for any of the groups constructed in response to Question 2? Question 1 was raised by Delzant and Gromov. Question 2 is intimately related to Question 1: the non-trivial examples of Kähler subgroups of direct products of surface groups never admit a classifying space with finite skeleton. The only known source of non-trivial examples for Questions 1 and 2 are fundamental groups of fibres of holomorphic maps from a direct product of closed surfaces onto an elliptic curve; the first such construction is due to Dimca, Papadima and Suciu. Question 3 was posed by Suciu in the context of these examples. In this thesis we: provide the first constraints on Kähler subdirect products of surface groups (Theorem 7.3.1); develop new construction methods for Kähler groups from maps onto higher-dimensional complex tori (Section 6.1); apply these methods to obtain irreducible examples of Kähler subgroups of direct products of surface groups which arise from maps onto higher-dimensional tori and use them to show that our conditions in Theorem 7.3.1 are minimal (Theorem A); apply our construction methods to produce irreducible examples of Kähler groups that (i) have a classifying space with finite (n-1)-skeleton but no classifying space with finite n-skeleton and (ii) do not have a subgroup of finite index which embeds in a direct product of surface groups (Theorem 8.3.1); provide a new proof of Biswas, Mj and Pancholi's generalisation of Dimca, Papadima and Suciu's construction to more general maps onto elliptic curves (Theorem 4.3.2) and introduce invariants that distinguish many of the groups obtained from this construction (Theorem 4.6.2); and, construct explicit finite presentations for Dimca, Papadima and Suciu's groups thereby answering Question 3 (Theorem 5.4.4)).
- Published
- 2017
18. Branched covers of elliptic curves and K\'ahler groups with exotic finiteness properties
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Isenrich, Claudio Llosa
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Mathematics - Geometric Topology ,Mathematics - Differential Geometry ,Mathematics - Group Theory ,32J27, 20J05, 20F65 - Abstract
We construct K\"ahler groups with arbitrary finiteness properties by mapping products of closed Riemann surfaces holomorphically onto an elliptic curve: for each $r\geq 3$, we obtain large classes of K\"ahler groups that have classifying spaces with finite $(r-1)$-skeleton but do not have classifying spaces with finitely many $r$-cells. We describe invariants which distinguish many of these groups. Our construction is inspired by examples of Dimca, Papadima and Suciu., Comment: 22 pages, 3 figures, V3: Minor corrections and improvements to the exposition. Final accepted version, to appear in the Annales de l'Institut Fourier
- Published
- 2016
19. Finite presentations for K\'ahler groups with arbitrary finiteness properties
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Isenrich, Claudio Llosa
- Subjects
Mathematics - Group Theory ,Mathematics - Differential Geometry ,Mathematics - Geometric Topology ,20F05, 20F34, 32J27 (Primary), 20J05 (Secondary) - Abstract
We construct the first explicit finite presentations for a family of K\"ahler groups with arbitrary finiteness properties, answering a question of Suciu., Comment: 19 pages, 1 figure. Final accepted version. To appear in the Journal of Algebra
- Published
- 2015
- Full Text
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20. Hyperbolic groups containing subgroups of type $\mathscr{F}_{3}$ not $\mathscr{F}_{4}$
- Author
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Py, Pierre, Isenrich, Claudio Llosa, Martelli, Bruno, Py, Pierre, Institut de Recherche Mathématique Avancée (IRMA), and Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)
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Mathematics - Geometric Topology ,hyperbolic groups ,FOS: Mathematics ,homological methods in group theory ,Geometric Topology (math.GT) ,topological methods in group theory ,[MATH] Mathematics [math] ,Group Theory (math.GR) ,[MATH]Mathematics [math] ,Mathematics - Group Theory ,20F67 (Primary), 20F65, 57M07, 20J05, 20F05 (Secondary) - Abstract
We give examples of hyperbolic groups which contain subgroups that are of type $\mathscr{F}_{3}$ but not of type $\mathscr{F}_{4}$. These groups are obtained by Dehn filling starting from a non-uniform lattice in ${\rm PO}(8,1)$ which was previously studied by Italiano, Martelli and Migliorini., 24 pages, 1 figure
- Published
- 2021
21. Residually free groups do not admit a uniform polynomial isoperimetric function.
- Author
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Isenrich, Claudio Llosa and Tessera, Romain
- Subjects
- *
FREE groups , *ISOPERIMETRIC inequalities , *POLYNOMIALS , *GROUP products (Mathematics) - Abstract
We show that there is no uniform polynomial isoperimetric function for finitely presented subgroups of direct products of free groups by producing a sequence of subgroups Gr ≤ F2(1) × ⋅⋅⋅ × F2(r) of direct products of 2-generated free groups with Dehn functions bounded below by nr. The groups Gr are obtained from the examples of non-coabelian subdirect products of free groups constructed by Bridson, Howie, Miller, and Short. As a consequence we obtain that residually free groups do not admit a uniform polynomial isoperimetric function. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
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22. KODAIRA FIBRATIONS, KÄHLER GROUPS, AND FINITENESS PROPERTIES.
- Author
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BRIDSON, MARTIN R. and ISENRICH, CLAUDIO LLOSA
- Subjects
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HOLOMORPHIC functions , *FINITE, The , *ELLIPTIC curves , *GROUP products (Mathematics) , *FIBERS - Abstract
We construct classes of Kähler groups that do not have finite classifying spaces and are not commensurable to subdirect products of surface groups. Each of these groups is the fundamental group of the generic fibre of a holomorphic map from a product of Kodaira fibrations onto an elliptic curve. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
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