Publications and Preprints
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Automorphisms of the Artin group of type D5.[ arXiv · ]For the Artin group of type D5, we determine its automorphism group and the automorphism group of its quotient by the center. This settles the only remaining case, n = 5, in the classification of automorphisms of Artin groups of spherical type Dn.
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Property R∞ for generalized Higman groups.[ arXiv · ]We give a unified proof of property R∞ for the Higman groups Hn (n ≥ 4) and for their generalizations studied by Martin and Horbez–Huang. As a key step, we prove that the automorphism groups of these groups are acylindrically hyperbolic. As a byproduct, we obtain acylindrical hyperbolicity of the groups themselves. In addition, we give an independent proof, based on Delzant's lemma, of a criterion implying property R∞.
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Humanity's Last Exam.Humanity's Last Exam is a multi-modal benchmark at the frontier of human knowledge, designed to evaluate advanced AI systems on expert-level questions across many subjects. The benchmark uses closed-ended questions suitable for automated grading and exposes a substantial gap between current model performance and the expert human frontier.
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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 n96, giving 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 Brady's cube complex with respect to the 4-point condition.
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Property R∞ for new classes of Artin groups.We establish property R∞ for Artin groups of spherical type Dn, n ≥ 6, their central quotients, large hyperbolic-type free-of-infinity Artin groups, and some other classes of large-type Artin groups. The key ingredients are descriptions of the automorphism groups of these Artin groups and their actions on suitable Gromov-hyperbolic spaces. We also give a detailed proof of Delzant's Lemma.
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Endomorphisms of Artin groups of type Bn.We determine a classification of the endomorphisms of the Artin groups of spherical type Bn for n ≥ 5, and of their quotients by the center.
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Endomorphisms of Artin groups of type Ãn.We determine a classification of the endomorphisms of the Artin group of affine type Ãn for n ≥ 4.
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Divergence, thickness and hypergraph index for general Coxeter groups.We study divergence and thickness for general Coxeter groups. We characterize linear divergence, show that superlinear divergence is at least quadratic, and define a computable hypergraph index for arbitrary Coxeter systems. Finite hypergraph index gives polynomial upper bounds on divergence and bounds on the order of thickness; we prove sharpness for certain families and obtain a general upper bound in terms of the topology of the associated Dynkin diagram.
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Property R∞ for some spherical and affine Artin-Tits groups.For n ≥ 2, we give a short uniform proof of property R∞ for Artin–Tits groups of spherical types An, Bn, D4, I2(m), their pure subgroups, and Artin–Tits groups of affine types Ãn−1 and C̃n. In particular, this gives an alternative proof of property R∞ for pure Artin braid groups.
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Homological Dehn functions of groups of type FP2.We prove foundational results for homological Dehn functions of groups of type FP2, including superadditivity and invariance under quasi-isometry. We then study the homological Dehn functions of Leary's groups GL(S), obtaining methods for constructing uncountably many groups with a prescribed homological Dehn function. As an application, we obtain a group of type FP2 with quartic homological Dehn function and unsolvable word problem.
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Artin groups of types F4 and H4 are not commensurable with that of type D4.We resolve two previously undecided commensurability cases by showing that the Artin groups of types F4 and H4 are not commensurable with the Artin group of type D4. As a key step, we identify the abstract commensurator of the D4 Artin group with an extended mapping class group, determine its automorphism group, and describe torsion in irreducible spherical Artin groups modulo their centers.
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Linearity of some low-complexity mapping class groups.A longer earlier version with an alternative computation from first principles: [ pdf ]By analyzing presentations of pure mapping class groups of orientable surfaces, we identify several low-complexity cases with groups related to braid groups and the Artin group of type D4. As a corollary, we prove linearity of the pure mapping class groups in these cases.
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Realizable ranks of joins and intersections of subgroups in free groups.We study which pairs of ranks of H ∨ K and H ∩ K can occur for subgroups H and K of a non-abelian free group. Using the combinatorics of pushouts of core graphs, we obtain restrictions on realizable rank pairs, settle the remaining case m = 4 of Guzman's Group-Theoretic Conjecture, and prove the proposed description of the realizable locus when rank(H) = 2.
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Uncountably many quasi-isometry classes of groups of type FP.We show that previously constructed uncountable families of groups of type FP and of n-dimensional Poincaré duality groups contain uncountably many quasi-isometry classes. In particular, for every n ≥ 4 there are uncountably many quasi-isometry classes of acyclic n-manifolds admitting free, cocompact, properly discontinuous discrete group actions.
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Genus bounds in right-angled Artin groups.For a right-angled Artin group whose defining graph has chromatic number k, we prove that every non-trivial element has stable commutator length at least 1/(6k). If the defining graph is triangle-free, every non-trivial element has stable commutator length at least 1/20. The proofs use an elementary geometric argument inspired by earlier work of Culler.
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Dehn functions of subgroups of right-angled Artin groups.For every positive integer k, we construct right-angled Artin groups containing free-by-cyclic subgroups whose monodromy automorphisms grow as nk. As a consequence, we obtain finitely presented subgroups of right-angled Artin groups whose Dehn functions grow as nk+2.
Other texts
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Linearity criteria for poly-free groups[ pdf · ]In his article on what is now called Lubotzky's linearity criterion, Alexander Lubotzky established a criterion for a group Aut(F) to be linear, where F is a free group of finite rank, in terms of p-congruence systems. We generalize this result to groups of the form A semidirect F, where A is a finitely generated subgroup of Aut(F).
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On a question of Peter Sarnak[ pdf · ]In his 2012 MSRI Notes on thin groups, Peter Sarnak asks whether a specific pair of symplectic matrices generates an infinite-index subgroup in Sp(4,Z). We approach this question with a technique adapted from mapping class groups.