Article
Mathematics
Simon Andre
Summary: We prove that two co-Hopfian finitely generated virtually free groups are isomorphic if and only if they are elementarily equivalent. We also show that co-Hopfian finitely generated virtually free groups are homogeneous in the sense of model theory.
BULLETIN OF THE LONDON MATHEMATICAL SOCIETY
(2022)
Article
Mathematics
Z. Sela
Summary: The JSJ decomposition encodes automorphisms and virtually cyclic splittings of a hyperbolic group. However, for general finitely presented groups, it only encodes their splittings. In this sequence of papers, we focus on studying the automorphisms of hierarchically hyperbolic groups satisfying certain weak acylindricity conditions. To do so, we develop a higher rank JSJ decomposition, demonstrated in the case of a right angled Artin group in the first paper. Additionally, for studying automorphisms of a general HHG, we construct what we refer to as a higher rank Makanin-Razborov diagram, which serves as the initial step in building the higher rank JSJ.
GEOMETRIC AND FUNCTIONAL ANALYSIS
(2023)
Article
Mathematics
Arseniy (Senia) Sheydvasser
Summary: This paper considers three families of groups: the Bianchi groups SL(2, O) where O is the ring of integers of an imaginary, quadratic field; the groups SL double dagger(2, O) where O is a double dagger-order of a definite, rational quaternion algebra with an orthogonal involution; and the groups SL(2, O) where O is an order of a definite, rational quaternion algebra. It is shown that these groups are generated by elementary matrices if and only if O is semi-Euclidean (or double dagger-semi-Euclidean), which is a generalization of the usual notion of a Euclidean ring. The proofs are surprisingly simple and proceed by considering fundamental domains of Kleinian groups.
COMMUNICATIONS IN ALGEBRA
(2023)
Article
Mathematics
Matias Carrasco, John M. Mackay
Summary: In this study, we investigate the conformal dimension of the boundary at infinity of Gromov hyperbolic groups and characterize when it is attained. We are also able to determine which Gromov hyperbolic groups have conformal dimension 1, answering a question posed by Bonk and Kleiner.
INVENTIONES MATHEMATICAE
(2022)
Article
Mathematics
Bernhard Muhlherr, Gianluca Paolini, Saharon Shelah
Summary: We establish the foundations of first-order model theory for Coxeter groups, characterizing the superstable Coxeter groups of finite rank and Coxeter groups that are domains. We then focus on definability questions in right-angled Coxeter groups and 2-spherical Coxeter groups, proving results related to elementary subgroups and the equivalence problem. Lastly, we explore the model theoretic applications of reflection length, proving the disconnectedness of affine Coxeter groups.
JOURNAL OF ALGEBRA
(2022)
Article
Mathematics
B. Jacobson
Summary: In this paper, a modification to the construction of elementary amenable lacunary hyperbolic groups is made to produce finitely generated elementary amenable groups which are mixed identity-free. As a result, locally finite p-groups which are mixed identity-free are also obtained as a byproduct.
COMMUNICATIONS IN ALGEBRA
(2021)
Article
Mathematics
Hao Liang
Summary: We prove a factorization theorem for Fuchsian groups similar to those proved by Agol and Liu for 3-manifold groups. As an application, we build Makanin-Razborov diagrams to parameterize the collection of all discrete representations from an arbitrary but fixed finitely generated group G to PSL(2,R). We define a new class of groups called PSL(2,R)-discrete limit groups and then use the factorization theorem to obtain useful information about this class of groups.
JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES
(2023)
Article
Mathematics
Yi Liu
Summary: This paper proves that for any orientable finite-volume hyperbolic 3-manifold, the profinite isomorphism type of the fundamental group uniquely determines the isomorphism type of the first integral cohomology, as marked with the Thurston norm and the fibered classes. Moreover, up to finite ambiguity, the profinite isomorphism type determines the isomorphism type of the fundamental group among the class of finitely generated 3-manifold groups.
INVENTIONES MATHEMATICAE
(2023)
Article
Mathematics
Simon Andre, Jonathan Fruchter
Summary: We generalize Merzlyakov's theorem about the first-order theory of non-abelian free groups to all acylindrically hyperbolic groups. As a consequence, we prove a conjecture stating that acylindrically hyperbolic groups have trivial positive theory, which further strengthens the research achievements in this field.
JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES
(2022)
Article
Mathematics
Jone Lopez De Gamiz Zearra
Summary: This paper generalizes the results of Bridson, Howie, Miller and Short to limit groups over Droms right-angled Artin groups. It characterizes finitely presented residually free groups and solves the generalized conjugacy problem for finitely presented groups that are residually a Droms RAAG, showing that their finitely presentable subgroups are separable.
ALGEBRAIC AND GEOMETRIC TOPOLOGY
(2022)
Article
Mathematics, Applied
Yanyan Gao, Yangjiang Wei
Summary: This paper proposes group codes in the symmetric group algebras liqSn where liq is a finite field of characteristic q and Sn is a symmetric group of order n!. The authors compute the unique idempotents of liqSn corresponding to the characters of symmetric groups and use the results to characterize the minimum distances and dimensions of group codes. Furthermore, they construct MDS group codes and almost MDS group codes in liqS3 and liqS4.
Article
Mathematics
Simon Andre
Summary: The paper discusses elementary subgroups of finitely generated virtually free groups, and demonstrates that elementary subgroups of finitely generated free groups are free factors. It also provides an algorithm to determine if a subgroup generated by a finite subset is V3-elementary. Additionally, it proves that every elementary embedding of an equationally noetherian group into itself is an automorphism.
GROUPS GEOMETRY AND DYNAMICS
(2021)
Article
Mathematics, Applied
Youngju Kim
Summary: This article discusses non-elementary groups generated by two parabolic isometries, proving that the deformation space of a thrice-punctured sphere group acting on hyperbolic 4-space is 7-dimensional. Among them, there is a 5-dimensional parameter space of linked thrice-punctured sphere groups. In particular, there exists a 1-parameter family of discrete linked thrice-punctured sphere groups with fixed rotation angles of the two parabolic generators and their product.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2023)
Article
Mathematics
Sagar B. Kalane, John R. Parker
Summary: In this paper, we study a group generated by two unipotent parabolic elements of SU(2, 1) with distinct fixed points. We provide several conditions to ensure the discreteness and freeness of the group. Furthermore, we present a result on the diameter of a finite R-circle in the Heisenberg group.
MATHEMATISCHE ZEITSCHRIFT
(2023)
Article
Mathematics
Daniel Litt
Summary: The paper analyzes the dynamics of Galois action on the deformation rings of mod l representations of the geometric fundamental group of X, and proves several finiteness results for function fields over algebraically closed fields in arbitrary characteristic, along with a weak variant of the Frey-Mazur conjecture for function fields in characteristic 0. For example, it shows that the set of representations of pi(1) (X-an) into GL(n) (L) stemming from geometry is finite if X is a normal, connected variety over C.
DUKE MATHEMATICAL JOURNAL
(2021)
