Chloé Perin
We consider embeddings in a torsion-free hyperbolic group which are elementary in the sense of first-order logic. We give a description of these embeddings in terms of Sela's hyperbolic towers. We deduce as a corollary that subgroups elementarily embedded in finitely generated free groups are free factors.
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We give a complete characterization of torsion-free hyperbolic groups which are homogeneous in the sense of first-order logic, in terms of the JSJ decompositions of their free factors.
Uniform small cancellation in torsion-free hyperbolic groups
该研究针对非初等无扭双曲群建立一致小消去定理，并将其应用于证明这类群的一阶理论不具有模型伴侣。
Let $Γ$ be a torsion-free hyperbolic group. We show that the set of solutions of any system of equations with one variable in $Γ$ is a finite union of points and cosets of centralizers if and only if any two-generator subgroup of $Γ$ is free.
We show that the universal theory of torsion groups is strongly contained in the universal theory of finite groups. This answers a question of Dyson. We also prove that the universal theory of some natural classes of torsion groups is undecidable. Finally we observe that the universal theory of the class of hyperbolic groups is undecidable and use this observation to construct a lacunary hyperbolic group with undecidable universal theory. Surprisingly, torsion groups play an important role in the proof of the latter results.
We prove that the generic type of a non-cyclic torsion-free hyperbolic group G is foreign to any interpretable abelian group, hence also to any interpretable field. This result depends, among other things, on the definable simplicity of a non-cyclic torsion-free hyperbolic group, and we take the opportunity to give a proof of the latter using Sela's description of imaginaries in torsion-free hyperbolic groups. We also use the description of imaginaries to prove that if F is a free group of rank > 2 then no orbit of a finite tuple from F under Aut(F) is definable.
The following properties are preserved under elementary equivalence, among finitely generated groups: being hyperbolic (possibly with torsion), being hyperbolic and cubulable, and being a subgroup of a hyperbolic group. In other words, if a finitely generated group G has the same first-order theory as a group possessing one of the previous property, then G enjoys this property as well.
We use a recent result of Alexander and Nishinaka to show that if $G$ is a non-elementary torsion-free hyperbolic group and $R$ is a countable domain, then the group ring $RG$ is primitive. This implies that the group ring $KG$ of any non-elementary torsion-free hyperbolic group $G$ over a field $K$ is primitive.
In a remarkable series of papers, Zlil Sela classified the first-order theories of free groups and torsion-free hyperbolic groups using geometric structures he called towers. It was later proved by Chloé Perin that if $H$ is an elementarily embedded subgroup (or elementary submodel) of a torsion-free hyperbolic group $G$, then $G$ is a tower over $H$. We prove a generalization of Perin's result to toral relatively hyperbolic groups using JSJ and shortening techniques.
We give an example of a definable set in every free or torsion-free (non-elementary) hyperbolic group that is not in the Boolean algebra of equational sets. Hence, the theories of free and torsion-free (non-elementary) hyperbolic groups are not equational in the sense of G. Srour.
Let G be a torsion free hyperbolic group. We prove that the elementary theory of G is decidable and admits an effective quantifier elimination to boolean combination of AE-formulas. The existence of such quantifier elimination was previously proved by Sela.
We study to what extent torsion-free (Gromov)-hyperbolic groups are elementarily equivalent to their finite index subgroups. In particular, we prove that a hyperbolic limit group either is a free product of cyclic groups and surface groups, or admits infinitely many subgroups of finite index which are pairwise non elementarily equivalent.
We construct cocompact lattices in a product of trees which are not virtually torsion-free. This gives the first examples of hierarchically hyperbolic groups which are not virtually torsion-free
We prove that in a torsion-free hyperbolic group, an element is a test element if and only if it is not contained in a proper retract.
This paper is devoted to the first-order theory of torsion-free hyperbolic groups. One of its purposes is to review some results and to provide precise and correct statements and definitions, as well as some proofs and new results. A key concept is that of a tower (Sela) or NTQ system (Kharlampovich-Myasnikov). We discuss them thoroughly. We state and prove a new general theorem which unifies several results in the literature: elementarily equivalent torsion-free hyperbolic groups have isomorphic cores (Sela); if $H$ is elementarily embedded in a torsion-free hyperbolic group $G$, then $G$ is …
We show that any nonabelian free group $F$ of finite rank is homogeneous; that is for any tuples $\bar a$, $\bar b \in F^n$, having the same complete $n$-type, there exists an automorphism of $F$ which sends $\bar a$ to $\bar b$. We further study existential types and we show that for any tuples $\bar a, \bar b \in F^n$, if $\bar a$ and $\bar b$ have the same existential $n$-type, then either $\bar a$ has the same existential type as a power of a primitive element, or there exists an existentially closed subgroup $E(\bar a)$ (resp. $E(\bar b)$) of $F$ containing $\bar a$ (resp. $\bar b$) and a…
In 2006 Z. Sela and independently O. Kharlampovich and A. Myasnikov gave a solution to the Tarski problems by showing that two non-abelian free groups have the same elementary theory. Subsequently Z. Sela generalized the techniques used in his proof of the Tarski conjecture to classify all finitely generated groups elementary equivalent to a given torsion-free hyperbolic group. One important step in his analysis of the elementary theory of free and torsion-free hyperbolic groups is the Generalized Merzlyakov's Theorem. In our work we show that given a hyperbolic group $Γ$ and $Γ$-limit group $…
We proved that non-elementary discrete convergence groups are acylindrically hyperbolic.
We show that, given a finitely generated group $G$ as the coordinate group of a finite system of equations over a torsion-free hyperbolic group $Γ$, there is an algorithm which constructs a cover of a canonical solution diagram. The diagram encodes all homomorphisms from $G$ to $Γ$ as compositions of factorizations through $Γ$-NTQ groups and canonical automorphisms of the corresponding NTQ-subgroups. We also give another characterization of $Γ$-limit groups as iterated generalized doubles over $Γ$.
This paper describes some generalizations of the results presented in the book "Geometry of defining Relations in Groups" , of A.Yu.Ol'shanskii to the case of non-cyclic torsion-free hyperbolic groups. In particular, it is proved that for every non-cyclic torsion-free hyperbolic group, there exists a non-Abelian torsion-free quotient group in which all proper subgroups are cyclic, and the intersection of any two of them is not trivial.
We give an explicit construction of a maximal torsion-free finite-index subgroup of a certain type of Coxeter group. The subgroup is constructed as the fundamental group of a finite and non-positively curved polygonal complex. First we consider the special case where the universal cover of this polygonal complex is a hyperbolic building, and we construct finite-index embeddings of the fundamental group into certain cocompact lattices of the building. We show that in this special case the fundamental group is an amalgam of surface groups over free groups. We then consider the general case, and …
An $\ell^2$ Obstruction for Elementary Embeddings of Hyperbolic Groups
本文研究了超几何群在第一阶逻辑下嵌入时 $\ell^2$ 零数的单调性，证明了非阿贝尔proper子群在无 torsion 超几何群中的元素嵌入下严格递增，并进一步展示了任意有限生成群的存在性嵌入的单调性。
Let $G$ be a group and $g$ a non-trivial element in $G$. If some non-empty finite product of conjugates of $g$ equals to the trivial element, then $g$ is called a generalized torsion element. To the best of our knowledge, we have no hyperbolic $3$--manifold groups with generalized torsion elements whose rank is explicitly known to be greater than two. The aim of this short note is to demonstrate that for a given integer $n > 1$ there are infinitely many closed hyperbolic $3$--manifolds $M_n$ which enjoy the property: (i) the Heegaard genus of $M_n$ is $n$, (ii) the rank of the fundamental grou…
Let $G$ be a group hyperbolic relative to a collection of subgroups $\{H_λ,λ\in Λ\} $. We say that a subgroup $Q\le G$ is hyperbolically embedded into $G$, if $G$ is hyperbolic relative to $\{H_λ,λ\in Λ\} \cup \{Q\} $. In this paper we obtain a characterization of hyperbolically embedded subgroups. In particular, we show that if an element $g\in G$ has infinite order and is not conjugate to an element of $H_λ$, $λ\in Λ$, then the (unique) maximal elementary subgroup contained $g$ is hyperbolically embedded into $G$. This allows to prove that if $G$ is boundedly generated, then $G$ is elementar…
We construct an uncountable sequence of groups acting uniformly properly on hyperbolic spaces. We show that only countably many of these groups can be virtually torsion-free. This gives new examples of groups acting uniformly properly on hyperbolic spaces that are not virtually torsion-free and cannot be subgroups of hyperbolic groups.
In this note we prove the claim given in the title. A group G is noncommutatively slender if each map from the fundamental group of the Hawaiian Earring to G factors through projection to a canonical free subgroup. Graham Higman, in his seminal 1952 paper, proved that free groups are noncommutatively slender. Such groups were first defined by K. Eda. Eda has asked which finitely presented groups are noncommutatively slender. This result demonstrates that random finitely presented groups in the few- relator sense of Gromov are noncommutatively slender.
We study sets of solutions to equations over a free group, projections of such sets, and the structure of elementary sets defined over a free group. The structre theory we obtain enable us to answer some questions of A. Tarski's, and classify those finitely generated groups that are elementary equivalent to a free group. Connections with low dimensional topology, and a generalization to (Gromov) hyperbolic groups will also be discussed.
This paper proves that in a non-elementary relatively hyperbolic group, the logarithm growth rate of any non-elementary subgroup has a linear lower bound by the logarithm of the size of the corresponding generating set. As a consequence, any non-elementary subgroup has uniform exponential growth.
Let $H$ be a torsion-free $δ$-hyperbolic group with respect to a finite generating set $S$. Let $a_1,..., a_n$ and $a_{1*},..., a_{n*}$ be elements of $H$ such that $a_{i*}$ is conjugate to $a_i$ for each $i=1,..., n$. Then, there is a uniform conjugator if and only if $W(a_{1*},..., a_{n*})$ is conjugate to $W(a_1,..., a_n)$ for every word $W$ in $n$ variables and length up to a computable constant depending only on $δ$, $\sharp{S}$ and $\sum_{i=1}^n |a_i|$. As a corollary, we deduce that there exists a computable constant $\mathcal{C}=\mathcal{C}(δ, \sharp S)$ such that, for any endomorphism…
Around first-order rigidity of Coxeter groups
本文研究Coxeter群的一阶刚性，证明在有限挠生成模型下，初等等价或AE等价的Coxeter群仍为Coxeter群，并建立偶数双曲Coxeter群的一阶挠刚性及初等等价蕴含同构。
We study the word and conjugacy problems in lacunary hyperbolic groups (briefly, LHG). In particular, we describe a necessary and sufficient condition for decidability of the word problem in LHG. Then, based on the graded small-cancellation theory of Olshanskii, we develop a general framework which allows us to construct lacunary hyperbolic groups with word and conjugacy problems highly controllable and flexible both in terms of computability and computational complexity. As an application, we show that for any recursively enumerable subset $\mathcal{L} \subseteq \mathcal{A}^*$, where $\mathca…
Sela introduced limit groups in his work on the Tarski problem, and showed that each limit group has a cyclic hierarchy. In this paper, a class of relatively hyperbolic groups, equipped with a hierarchy similar to the one for limit groups, is shown to be closed under quasi-isometry. Additionally, these groups share some of the properties of limit groups. In particular, groups quasi-isometric to limit groups are shown to be LERF and virtually toral relatively hyperbolic.
We construct and classify all groups, given by triangular presentations associated to the smallest thick generalized quadrangle, that act simply transitively on the vertices of hyperbolic triangular buildings of the smallest non-trivial thickness. Our classification shows 23 non-isomorphic torsion free groups (obtained in an earlier work) and 168 non-isomorphic torsion groups acting on one of two possible buildings with the smallest thick generalized quadrangle as the link of each vertex. In analogy with the Euclidean case, we find both torsion and torsion free groups acting on the same buildi…
Examples of non-tame abstract elementary classes of abelian groups
本文构造了非 tame 的交换群抽象初级类，展示了其在不同可数基数下的特性，回答了相关问题并提供了首个自然语言中的非 tame 抽象初级类例子。
We provide a solution to the isomorphism problem for torsion-free relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsion-free hyperbolic groups (Sela); and (ii) fully residually free groups (Bumagin, Kharlampovich and Miasnikov). We also give a solution to the homeomorphism problem for finite volume hyperbolic n-manifolds, for $n \ge 3$. In the course of the proof of the main result, we prove that a particular JSJ decomposition of a freely indecomposable torsion-free relatively hyperbolic group with abelian parabo…
We show that a construction by Aanderaa and Cohen used in their proof of the Higman Embedding Theorem preserves torsion length. We give a new construction showing that every finitely presented group is the quotient of some $C'(1/6)$ finitely presented group by the subgroup generated by its torsion elements. We use these results to show there is a finitely presented group with infinite torsion length which is $C'(1/6)$, and thus word-hyperbolic and virtually torsion-free.
Sela proved every torsion-free one-ended hyperbolic group is coHopfian. We prove that there exist torsion-free one-ended hyperbolic groups that are not commensurably coHopfian. In particular, we show that the fundamental group of every simple surface amalgam is not commensurably coHopfian.
We study conjugacy classes of solutions to systems of equations and inequations over torsion-free hyperbolic groups, and describe an algorithm to recognize whether or not there are finitely many conjugacy classes of solutions to such a system. The class of immutable subgroups of hyperbolic groups is introduced, which is fundamental to the study of equations in this context. We apply our results to enumerate the immutable subgroups of a torsion-free hyperbolic group.
Reflective hyperbolic 2-elementary lattices, K3 surfaces and hyperkahler varieties
本文计算大反射偶2-初等双曲格的Coxeter图及其极大抛物子图，并将其应用于K3曲面和超凯勒簇理论。
Hyperbolic hyperbolic-by-cyclic groups are cubulable
本文证明双曲群经双曲自同构所得映射环面可立方化，并重新证明双曲 free-by-cyclic 群结果，同时把相关双曲性判据推广到带挠元情形。
We obtain a number of finiteness results for groups acting on Gromov-hyperbolic spaces. In particular we show that a torsion-free locally quasiconvex hyperbolic group has only finitely many conjugacy classes of $n$-generated one-ended subgroups. We also show that the rank problem is solvable for the class of torsion-free locally quasiconvex hyperbolic groups (even though it is unsolvable for the class of all torsion-free hyperbolic groups). We apply our results to 3-manifold groups. Namely, suppose $G$ is the fundamental group of a closed hyperbolic 3-manifold fibering over a circle and suppos…
First-order theory of torsion-free Tarski monsters
研究构造了无 torsion 自由塔斯基怪物，证明其与自由群有相同正理论，同时解决了随机商群中的单量词骑士猜想。
We generalize some results of Paulin and Rips-Sela on endomorphisms of hyperbolic groups to relatively hyperbolic groups, and in particular prove the following. (1) If G is a finitely generated non-elementary relatively hyperbolic group with slender parabolic subgroups, and either G is not co-Hopfian or Out(G) is infinite, then G splits over a slender group. (2) If a finitely generated non-parabolic subgroup H of a non-elementary relatively hyperbolic group is not Hopfian, then H acts non-trivially on an R-tree. (3) Every non-elementary relatively hyperbolic group has a non-elementary relative…
In this paper we exhibit Morse geodesics, often called "hyperbolic directions", in infinite unbounded torsion groups. The groups studied are lacunary hyperbolic groups and constructed using graded small cancellation conditions. In all previously known examples, Morse geodesics were found in groups which also contained Morse elements, infinite order elements whose cyclic subgroup gives a Morse quasi-geodesic. Our result presents the first example of a group which contains Morse geodesics but no Morse elements. In fact, we show that there is an isometrically embedded $7$-regular tree inside such…
We construct a 2-generator recursively presented group with infinite torsion length. We also explore the construction in the context of solvable and word-hyperbolic groups.
We study the (Ahlfors regular) conformal dimension of the boundary at infinity of Gromov hyperbolic groups which split over elementary subgroups. If such a group is not virtually free, we show that the conformal dimension is equal to the maximal value of the conformal dimension of the vertex groups, or 1, whichever is greater, and we characterise when the conformal dimension is attained. As a consequence, we are able to characterise which Gromov hyperbolic groups (without $2$-torsion) have conformal dimension 1, answering a question of Bonk and Kleiner.
It is known that a bi-orderable group has no generalized torsion element, but the converse does not hold in general. We conjecture that the converse holds for the fundamental groups of 3-manifolds, and verify the conjecture for non-hyperbolic, geometric 3-manifolds. We also confirm the conjecture for some infinite families of closed hyperbolic 3-manifolds. In the course of the proof, we prove that each standard generator of the Fibonacci group F(2,m) (m>2) is a generalized torsion element.
Quadratic Isoperimetric Inequality for Mapping Tori of Hyperbolic Groups
该研究确立了一端无挠双曲群的映射环面具有二次等周不等式，推进了几何群论相关问题的研究。
We show that the topological complexity of a finitely generated torsion free hyperbolic group $π$ with $\cdπ=n$ equals $2n$.
$L^2$-torsion of automorphisms
本文建立群自同构的$L^2$-挠率理论，证明相关组合公式，并将其应用于计算多类群、流形及相关自同构的$L^2$-挠率。
Relative cohomological dimension of a relatively hyperbolic pair
该研究证明无挠相对双曲对的相对上同调维数有限，且在特定条件下拟等距不变，并通过所提方法计算了多类情形的该维数。
For any finitely generated, non-elementary, torsion-free group $G$ that is hyperbolic relative to $\mathbb P$, we show that there exists a group $G^*$ containing $G$ such that $G^*$ is hyperbolic relative to $\mathbb P$ and $G$ is not relatively quasiconvex in $G^*$. This generalizes a result of I. Kapovich for hyperbolic groups. We also prove that any torsion-free group $G$ that is non-elementary and hyperbolic relative to $\mathbb P$, contains a rank 2 free subgroup $F$ such that the group generated by "randomly" chosen elements $r_1,...,r_m$ in $F$ is aparabolic, malnormal in $G$ and quasic…
We say that a group $G$ is acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space. We prove that the class of acylindrically hyperbolic groups coincides with many other classes studied in the literature, e.g., the class $C_{geom}$ introduced by Hamenstadt, the class of groups admitting a non-elementary weakly properly discontinuous action on a hyperbolic space in the sense of Bestvina and Fujiwara, and the class of groups with hyperbolically embedded subgroups studied by Dahmani, Guirardel, and the author. We also record some basic results about acyli…
We prove that the generalised word problem of a finitely generated subgroup of a finitely generated virtually free group is context-free, that a hyperbolic group must be virtually free if it has a torsion-free quasiconvex subgroup of infinite index with context-free generalised word problem, and that, for any hyperbolic group, the generalised word problem of a torsion-free quasiconvex subgroup is recognised by a real-time Turing machine.
We show that the full set of solutions to systems of equations and inequations in a hyperbolic group, with or without torsion, as shortlex geodesic words, is an EDT0L language whose specification can be computed in $\mathsf{NSPACE}(n^2\log n)$ for the torsion-free case and $\mathsf{NSPACE}(n^4\log n)$ in the torsion case. Our work combines deep geometric results by Rips, Sela, Dahmani and Guirardel on decidability of existential theories of hyperbolic groups, work of computer scientists including Plandowski, Jeż, Diekert and others on $\mathsf{PSPACE}$ algorithms to solve equations in free mon…
In a group, a non-trivial element is called a generalized torsion element if some non-empty finite product of its conjugates equals to the identity. We say that a knot has generalized torsion if its knot group admits such an element. For a (2, 2q+1)-torus knot K, we demonstrate that there are infinitely many unknots c such that p-twisting K about c yields a twist family, which consists of hyperbolic knots with generalized torsion whenever |p| > 3. This gives a new infinite class of hyperbolic knots having generalized torsion. In particular, each class contains knots with arbitrarily high genus…
This paper aims to give an account of theorem of Louder and Touikan which shows that many hierarchies consisting of slender JSJ-decompositions are finite. In particular JSJ-hierarchies of $2$-torsion-free hyperbolic groups are always finite.
In this paper we use JSJ-decompositions to formalise a folk conjecture recorded by Pride on the structure of one-relator groups with torsion. We prove a slightly weaker version of the conjecture, which implies that the structure of one-relator groups with torsion closely resemble the structure of torsion-free hyperbolic groups.
Building on previous results concerning hyperbolicity of groups of Fibonacci type, we give an almost complete classification of the (non-elementary) hyperbolic groups within this class. We are unable to determine the hyperbolicity status of precisely two groups, namely the Gilbert-Howie groups H(9,4), H(9,7). We show that if H(9,4) is torsion-free then it is not hyperbolic. We consider the class of T(5) cyclically presented groups and classify the (non-elementary) hyperbolic groups and show that the Tits alternative holds.
We exhibit the first examples of residually finite non-linear Gromov hyperbolic groups. Our examples are constructed as amalgamated products of torsion-free cocompact lattices in the rank 1 Lie group $\mathrm{Sp}(d,1)$, $d\geq 2$ along maximal cyclic subgroups.
Undecidability of epimorphisms onto products of hyperbolic groups
研究证明了某些超几何群积的满射不可判定性，通过构造递归序列的超几何群来展示算法无法判断商群同构性。
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