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For n ≥ 2 , let G 1 = A 1 ∗ ⋯ ∗ A n and G 2 = B 1 ∗ ⋯ ∗ B n where the A i 's and B i 's are non-elementary relatively hyperbolic groups. Suppose that, for 1 ≤ i ≤ n , the Bowditch boundary of A i is homeomorphic to the Bowditch boundary of B i . We show that the Bowditch boundary of G 1 is homeomorphic to the Bowditch boundary of G 2 . We generalize this result to graphs of relatively hyperbolic groups with finite edge groups. This extends Martin–Świątkowski's work in the context of relatively hyperbolic groups. [ABSTRACT FROM AUTHOR]

Let G$G$ be a residually finite, good group of finite virtual cohomological dimension. We prove that the natural monomorphism G↪Ĝ$G\hookrightarrow {\widehat{G}}$ induces a bijective correspondence between conjugacy classes of finite p$p$‐subgroups of G$G$ and those of its profinite completion Ĝ${\widehat{G}}$. Moreover, we prove that the centralizers and normalizers in Ĝ${\widehat{G}}$ of finite p$p$‐subgroups of G$G$ are the closures of the respective centralizers and normalizers in G$G$. With somewhat more restrictive hypotheses, we prove the same results for finite solvable subgroups of G$G$. In the last section, we give a few applications of this theorem to hyperelliptic mapping class groups and virtually compact special toral relatively hyperbolic groups (these include fundamental groups of 3‐orbifolds and of uniform standard arithmetic hyperbolic orbifolds). [ABSTRACT FROM AUTHOR]

Given a finite presentation for a group G which can also be realised as a relative presentation we give conditions on the relative presentation which, if satisfied, proves G hyperbolic. Using a curvature distribution method we confirm these conditions for the length four one-relator relative presentation 〈 u , t | t n , t m u t u − r 〉 for many values of r and n deduce that the corresponding generalised Fibonacci groups F (r , n) are hyperbolic. [ABSTRACT FROM AUTHOR]

We introduce a hyperbolic reflection group trick which builds closed aspherical manifolds out of compact ones and preserves hyperbolicity, residual finiteness, and—for almost all primes p— -homology growth above the middle dimension. We use this trick, embedding theory and manifold topology to construct Gromov hyperbolic 7-manifolds that do not virtually fiber over a circle out of graph products of large finite groups. [ABSTRACT FROM AUTHOR]

In this paper, we begin the systematic study of group equations with abelian predicates in the main classes of groups where solving equations is possible. We extend the line of work on word equations with length constraints, and more generally, on extensions of the existential theory of semigroups, to the world of groups. We use interpretability by equations to establish model-theoretic and algebraic conditions, which are sufficient to get undecidability. We apply our results to (non-abelian) right-angled Artin groups and show that the problem of solving equations with abelian predicates is undecidable for these. We obtain the same result for hyperbolic groups whose abelianisation has torsion-free rank at least two. By contrast, we prove that in groups with finite abelianisation, the problem can be reduced to solving equations with recognisable constraints, and so this is decidable in right-angled Coxeter groups, or more generally, graph products of finite groups, as well as hyperbolic groups with finite abelianisation. [ABSTRACT FROM AUTHOR]

We give a characterization for finite height subgroups in relatively hyperbolic groups by showing that a finitely generated, undistorted subgroup H of a relatively hyperbolic group (G, P) has finite height if and only H ∩ gPg-1 has finite height in gPg-1 for each conjugate of peripheral subgroup in P. Additionally, we prove that the concepts of finite height and strongly quasiconvexity are equivalent within the class of extended admissible groups. This class includes both the fundamental groups of non-geometric 3-mani-folds and Croke-Kleiner admissible groups. [ABSTRACT FROM AUTHOR]

The Sullivan dictionary provides a beautiful correspondence between Kleinian groups acting on hyperbolic space and rational maps of the extended complex plane. We focus on the setting of geometrically finite Kleinian groups with parabolic elements and parabolic rational maps. In this context an especially direct correspondence exists concerning the dimension theory of the associated limit sets and Julia sets. In recent work we established formulae for the Assouad type dimensions and spectra for these fractal sets and certain conformal measures they support. This allows a rather more nuanced comparison of the two families in the context of dimension. In this expository article we discuss how these results provide new entries in the Sullivan dictionary, as well as revealing striking differences between the two families. [ABSTRACT FROM AUTHOR]

Let Γ be a nonelementary hyperbolic group with a word metric d and ∂Γ its hyperbolic boundary equipped with a visual metric da for some parameter a>1. Fix a superexponential symmetric probability μ on Γ whose support generates Γ as a semigroup, and denote by ρ the spectral radius of the random walk Y on Γ with step distribution μ. Let ν be a probability on 1,2,3,... with mean λ=∑k=1∞kνk<∞. Let BRWΓνμ be the branching random walk on Γ with offspring distribution ν and base motion Y , and let Hλ be the volume growth rate for the trace of BRWΓνμ. We prove for λ∈1ρ−1 that the Hausdorff dimension of the limit set Λ , which is the random subset of ∂Γda consisting of all accumulation points of the trace of BRWΓνμ , is given by logaHλ. Furthermore, we prove that Hλ is almost surely a deterministic, strictly increasing, and continuous function of λ∈1ρ−1 , is bounded by the square root of the volume growth rate of Γ , and has critical exponent 1/2 at ρ−1 in the sense that Hρ−1−Hλ∼Cρ−1−λasλ↑ρ−1 for some positive constant C. We conjecture that the Hausdorff dimension of Λ in the critical case λ=ρ−1 is logaHρ−1 almost surely. This has been confirmed on free groups or the free product (by amalgamation) of finitely many finite groups equipped with the word metric d defined by the standard generating set. © 2022 Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

We show that groups presented by inverse-closed finite confluent length-reducing rewriting systems are characterised by a striking geometric property: their Cayley graphs are geodetic and side-lengths of non-degenerate geodesic triangles are uniformly bounded. This leads to a new algebraic result: the group is plain (isomorphic to the free product of finitely many finite groups and copies of Z) if and only if a certain relation on the set of non-trivial finite-order elements of the group is transitive on a bounded set. We use this to prove that deciding if a group presented by an inverse-closed finite confluent length-reducing rewriting system is not plain is in NP. A 'yes' answer to an instance of this problem would disprove a longstanding conjecture of Madlener and Otto from 1987. We also prove that the isomorphism problem for plain groups presented by inverse-closed finite confluent length-reducing rewriting systems is in PSPACE. • A new geometric characterisation of groups presented by inverse-closed finite confluent length-reducing rewriting systems. • A new algebraic characterisation of groups presented by inverse-closed finite confluent length-reducing rewriting systems. • Deciding if a group presented by inverse-closed finite confluent length-reducing rewriting systems is not plain is in NP. • Deciding isomorphism between plain groups given by inverse-closed finite confluent length-reducing rewriting systems is in PSPACE. [ABSTRACT FROM AUTHOR]

We study unions of fundamental domains of a Fuchsian group, especially those with hyperbolic plane metric realizing the metric of the corresponding hyperbolic surface. We call these unions the geodesic covers of the Fuchsian group or the hyperbolic surface. The paper contributes to showing that finiteness of geodesic covers is basically another characterization of geometrically finiteness. The resolution of geometrically finite case is based on Shimizu's lemma. [ABSTRACT FROM AUTHOR]

We propose several common extensions of the classes of Anosov subgroups and geometrically finite Kleinian groups among discrete subgroups of semisimple Lie groups. We relativize various dynamical and coarse geometric characterizations of Anosov subgroups given in our earlier work, extending the class from intrinsically hyperbolic to relatively hyperbolic subgroups. We prove implications and equivalences between the various relativizations. [ABSTRACT FROM AUTHOR]

We show that the class of groups where EDT0L languages can be used to describe solution sets to systems of equations is closed under direct products, wreath products with finite groups, and passing to finite index subgroups. We also add the class of groups that contain a direct product of hyperbolic groups as a finite index subgroup to the list of groups where solutions to systems of equations can be expressed as an EDT0L language. This includes dihedral Artin groups. We also show that the systems of equations with rational constraints in virtually abelian groups have EDT0L solutions, and the addition of recognisable contraints to any system preserves the property of having EDT0L solutions. These EDT0L solutions are expressed with respect to quasigeodesic normal forms. We discuss the space complexity in which EDT0L systems for these languages can be constructed. [ABSTRACT FROM AUTHOR]

For any orientable finite-volume hyperbolic 3-manifold, this paper proves that 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. [ABSTRACT FROM AUTHOR]

Bestvina [1] introduced the notion of a (weak) Z -structure and (weak) Z -boundary for a torsion-free group, motivated by the notion of boundary for hyperbolic and C A T (0) groups. Since then, some classes of groups have been shown to admit a (weak) Z -structure (see [5,20,22] for example); in fact, in all cases these groups are semistable at infinity and happen to have a pro-(finitely generated free) fundamental pro-group. The question whether or not every type F group admits such a structure remains open. In [33] it was shown that the property of admitting such a structure is closed under direct products and free products. Our main results are as follows. THEOREM: Let G be a torsion-free and semistable at infinity finitely presented group with a pro-(finitely generated free) fundamental pro-group at each end. If G has a finite graph of groups decomposition in which all the groups involved are of type F and inward tame (in particular, if they all admit a weak Z -structure) then G admits a weak Z -structure. COROLLARY: The class of those 1-ended and semistable at infinity torsion-free finitely presented groups which admit a weak Z -structure and have a pro-(finitely generated free) fundamental pro-group is closed under amalgamated products (resp. HNN-extensions) over finitely generated free groups. On the other hand, given a finitely presented group G and a monomorphism φ : G ⟶ G , we may consider the ascending HNN-extension G ⁎ φ = 〈 G , t ; t − 1 g t = φ (g) , g ∈ G 〉. The results in [26] together with the Theorem above yield the following: PROPOSITION: If a finitely presented torsion-free group G is of type F and inward tame, then any (1-ended) ascending HNN-extension G ⁎ φ admits a weak Z -structure. In the particular case φ ∈ A u t (G) , this ascending HNN-extension corresponds to a semidirect product G ⋊ φ Z , and it has been shown in [18] that if G admits a Z -structure then so does G ⋊ φ Z. [ABSTRACT FROM AUTHOR]

The main purposes of the present paper are to introduce the notion of a hyperbolic curvoid and to study the geometry of hyperbolic curvoids. A hyperbolic curvoid is defined to be a certain profinite group and may be considered to be "group-theoretic abstraction" of the notion of a hyperbolic curve from the viewpoint of anabelian geometry. One typical example of a hyperbolic curvoid is a profinite group isomorphic to the 'etale fundamental group of a hyperbolic curve either over a number field or over a mixed-characteristic nonarchimedean local field. The first part of the paper centers around the establishment of a construction algorithm for the "geometric subgroup" of hyperbolic curvoids and a construction algorithm for the "collection of cuspidal inertia subgroups" of hyperbolic curvoids. Moreover, we also consider respective analogues for hyperbolic curvoids of the theory of partial compactifications of hyperbolic curves and the theory of quotient orbicurves of hyperbolic curves by actions of finite groups. [ABSTRACT FROM AUTHOR]

We show that the automorphism groups of right-angled Artin groups whose defining graphs have at least three vertices are not relatively hyperbolic. We then show that the outer automorphism groups are also not relatively hyperbolic, except for a few exceptional cases. In these cases, the outer automorphism groups are virtually isomorphic to either a finite group, an infinite cyclic group or $\mathrm {GL}_2(\mathbb {Z})$. [ABSTRACT FROM AUTHOR]

In this paper, we derive an asymptotic formula for the number of conjugacy classes of elements in a class of statistically convex-cocompact actions with contracting elements. Denote by C(o, n) (respectively, C'(o, n)) the set of (respectively, primitive) conjugacy classes of algebraic length at most n for a basepoint o. The main result is the following asymptotic formula: ... A similar formula holds for conjugacy classes using stable length. As a consequence of the formulae, the conjugacy growth series is transcendental for all nonelementary relatively hyperbolic groups, graphical small cancellation groups with finite components. As a byproduct of the proof, we establish several useful properties for an exponentially generic set of elements. In particular, it yields a positive answer to a question of J. Maher that an exponentially generic element in mapping class groups has their Teichmüller axis contained in the principal stratum. [ABSTRACT FROM AUTHOR]

Nikulin [On the quotient groups of the automorphism groups of hyperbolic forms by the subgroups generated by 2 reflections, J. Soviet. Math.22 (1983) 1401–1476; Surfaces of type K 3 with a finite automorphism group and a Picard group of rank three, Proc. Steklov Institute of Math. (3) (1985) 131–155] and Vinberg [Classification of 2-reflective hyperbolic lattices of rank 4, Trans. Moscow Math. Soc. (2007) 39–66] proved that there are only a finite number of lattices of rank ≥ 3 that are the Néron–Severi lattice of projective K 3 surfaces with a finite automorphism group. The aim of this paper is to provide a more geometric description of such K 3 surfaces X , when these surfaces have moreover no elliptic fibrations. In that case, we show that such K 3 surface is either a quartic with special hyperplane sections or a double cover of the plane branched over a smooth sextic curve which has special tangencies properties with some lines, conics or cuspidal cubic curves. We then study the converse, i.e. if the geometric description we obtained characterizes these surfaces. In four cases, the description is sufficient, in each of the four other cases, there is exactly another one possibility which we study. We obtain that at least five moduli spaces of K 3 surfaces (among the eight we study) are unirational. [ABSTRACT FROM AUTHOR]

We show that any graph product of finitely generated groups is hierarchically hyperbolic relative to its vertex groups. We apply this result to answer two questions of Behrstock, Hagen, and Sisto: we show that the syllable metric on any graph product forms a hierarchically hyperbolic space, and that graph products of hierarchically hyperbolic groups are themselves hierarchically hyperbolic groups. This last result is a strengthening of a result of Berlai and Robbio by removing the need for extra hypotheses on the vertex groups. We also answer two questions of Genevois about the geometry of the electrification of a graph product of finite groups. [ABSTRACT FROM AUTHOR]

Let G be a finitely presented group, and let H be a subgroup of G. We prove that if H is acylindrically hyperbolic and existentially closed in G, then G is acylindrically hyperbolic. As a corollary, any finitely presented group which is existentially equivalent to the mapping class group of a surface of finite type, to \mathrm {Out}(F_n) or \mathrm {Aut}(F_n) for n\geq 2 or to the Higman group, is acylindrically hyperbolic. [ABSTRACT FROM AUTHOR]

Given a finite subgroup G of the mapping class group of a surface S, the Nielsen realization problem asks whether G can be realized as a finite group of homeomorphisms of S. In 1983, Kerckhoff showed that for S a finite-type surface, any finite subgroup G may be realized as a group of isometries of some hyperbolic metric on S. We extend Kerckhoff's result to orientable, infinite-type surfaces. As applications, we classify torsion elements in the mapping class group of the plane minus a Cantor set, and also show that topological groups containing sequences of torsion elements limiting to the identity do not embed continuously into the mapping class group of S. Finally, we show that compact subgroups of the mapping class group of S are finite, and locally compact subgroups are discrete. [ABSTRACT FROM AUTHOR]

Let M be a closed oriented 3-manifold and let G be a discrete group. We consider a representation ρ : π 1 (M) → G. For a 3-cocycle α , the Dijkgraaf–Witten invariant is given by (ρ ∗ α) [ M ] , where ρ ∗ : H 3 (G) → H 3 (M) is the map induced by ρ , and [ M ] denotes the fundamental class of M. Note that (ρ ∗ α) [ M ] = α (ρ ∗ [ M ]) , where ρ ∗ : H 3 (M) → H 3 (G) is the map induced by ρ , we consider an equivalent invariant ρ ∗ [ M ] ∈ H 3 (G) , and we also regard it as the Dijkgraaf–Witten invariant. In 2004, Neumann described the complex hyperbolic volume of M in terms of the image of the Dijkgraaf–Witten invariant for G = SL 2 ℂ by the Bloch–Wigner map from H 3 ( SL 2 ℂ) to the Bloch group of ℂ. In this paper, by replacing ℂ with a finite field 𝔽 p , we calculate the reduced Dijkgraaf–Witten invariants of the complements of twist knots, where the reduced Dijkgraaf–Witten invariant is the image of the Dijkgraaf–Witten invariant for SL 2 𝔽 p by the Bloch–Wigner map from H 3 ( SL 2 𝔽 p) to the Bloch group of 𝔽 p . [ABSTRACT FROM AUTHOR]

Let Γ be a relatively hyperbolic group and let μ be an admissible symmetric finitely supported probability measure on Γ . We extend Floyd–Ancona type inequalities from Gekhtman et al. (Martin boundary covers Floyd boundary, 2017. arXiv:1708.02133) up to the spectral radius R of μ . We use them to find the precise homeomorphism type of the r-Martin boundary, which describes r-harmonic functions, for every r ≤ R . We also define a notion of spectral degeneracy along parabolic subgroups which is crucial to describe the homeomorphism type of the R-Martin boundary. Finally, we give a criterion for (strong) stability of the Martin boundary in the sense of Picardello and Woess (in: Potential theory, de Gruyter, 1992) in terms of spectral degeneracy. We then prove that this criterion is always satisfied in small rank, so that in particular, the Martin boundary of an admissible symmetric finitely supported probability measure on a geometrically finite Kleinian group of dimension at most 5 is always strongly stable. [ABSTRACT FROM AUTHOR]

We propose a criterion for preserving the regularity of a formal language representation when passing from groups to subgroups. We use this criterion to show that the regularity of a positive cone language in a left-orderable group passes to its finite index subgroups, and to show that there exists no left order on a finitely generated acylindrically hyperbolic group such that the corresponding positive cone is represented by a quasi-geodesic regular language. We also answer one of Navas' questions by giving an example of an infinite family of groups which admit a positive cone that is generated by exactly k generators, for every k ≥ 3. As a special case of our construction, we obtain a finitely generated positive cone for F 2 × ℤ. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We use assembly maps to study 𝐓𝐂 ⁢ (𝔸 ⁢ [ G ] ; p) \mathbf{TC}(\mathbb{A}[G];p) , the topological cyclic homology at a prime p of the group algebra of a discrete group G with coefficients in a connective ring spectrum 𝔸 \mathbb{A}. For any finite group, we prove that the assembly map for the family of cyclic subgroups is an isomorphism on homotopy groups. For infinite groups, we establish pro-isomorphism, (split) injectivity, and rational injectivity results, as well as counterexamples to injectivity and surjectivity. In particular, for hyperbolic groups and for virtually finitely generated abelian groups, we show that the assembly map for the family of virtually cyclic subgroups is injective but in general not surjective. [ABSTRACT FROM AUTHOR]

In this paper, we prove a combination theorem for a complex of relatively hyperbolic groups. It is a generalization of Martin's (Geom. Topology 18 (2014), 31–102) work for combination of hyperbolic groups over a finite MK-simplicial complex, where k ≤ 0. [ABSTRACT FROM AUTHOR]

We prove that, for any two finite volume hyperbolic 3‐manifolds, the amalgamation of their fundamental groups along any nontrivial infinite index geometrically finite subgroup is not LERF (locally extended residually finite). This generalizes the author's previous work on nonLERFness of amalgamations of hyperbolic 3‐manifold groups along abelian subgroups. A consequence of this result is that, for all closed arithmetic hyperbolic 4‐manifolds have nonLERF fundamental groups. Along with the author's previous work, we obtain that, for any arithmetic hyperbolic manifold with dimension at least 4, with possible exceptions in seven‐dimensional manifolds defined by the octonion, its fundamental group is not LERF. [ABSTRACT FROM AUTHOR]

We formulate and prove a Bott periodicity theorem for an ℓ p -space (1 ≤ p < ∞). For a proper metric space X with bounded geometry, especially for a coarsely connected space, we introduce a version of K -homology at infinity and the Roe algebra at infinity and show that to prove the coarse Novikov conjecture, it suffices to prove the coarse assembly map at infinity is an injection. As a result, we show that the coarse Novikov conjecture holds for any metric space with bounded geometry which admits a fibred coarse embedding into an ℓ p -space. These include all box spaces of a residually finite hyperbolic group, and a large class of warped cones of a compact metric space with an action by a hyperbolic group. [ABSTRACT FROM AUTHOR]

Given non‐trivial finite groups A and B, not both of order 2, we prove that every finite simple group of sufficiently large rank is an image of the free product A*B. To show this, we prove that every finite simple group of sufficiently large rank is generated by a pair of subgroups isomorphic to A and B. This proves a conjecture of Tamburini and Wilson. [ABSTRACT FROM AUTHOR]

Let G be a locally finite group and let F(G) be the Hirsch–Plotkin radical of G. Let S denote the full inverse image of the generalized Fitting subgroup of G/F(G) in G. Assume that there is a number k such that the length of every nested chain of centralizers in G does not exceed k. The Borovik–Khukhro conjecture states, in particular, that under this assumption, the quotient G/S contains an abelian subgroup of finite index bounded in terms of k. We disprove this statement and prove a weak analogue of it. [ABSTRACT FROM AUTHOR]

A subgroup of a group G is called algebraic if it can be expressed as a finite union of solution sets to systems of equations. We prove that a non-elementary subgroup H of an acylindrically hyperbolic group G is algebraic if and only if there exists a finite subgroup K of G such that C G ( K ) ≤ H ≤ N G ( K ) . We provide some applications of this result to free products, torsion-free relatively hyperbolic groups, and ascending chains of algebraic subgroups in acylindrically hyperbolic groups. [ABSTRACT FROM AUTHOR]

We show, using acylindrical hyperbolicity, that a finitely generated group splitting over Z cannot be simple. We also obtain SQ-universality in most cases, for instance a balanced group (one where if two powers of an infinite order element are conjugate then they are equal or inverse) which is finitely generated and splits over Z must either be SQ-universal or it is one of exactly seven virtually abelian exceptions. [ABSTRACT FROM AUTHOR]

A theorem of E.C. Turner states that ifFis a finitely generated free group, then the test words are precisely the elements not contained in any proper retract. In this paper, we examine some ideas in model theory and logic related to Turner’s characterization of test words and introduce Turner groups, a class of groups containing all finite groups and all stably hyperbolic groups satisfying this characterization. We show that Turner’s theorem is not first-order expressible. However, we prove that every finitely generated elementary free group is a Turner group. [ABSTRACT FROM AUTHOR]

We prove that there is no algorithm that can determine whether or not a finitely presented group has a non-trivial finite quotient; indeed, this property remains undecidable among the fundamental groups of compact, non-positively curved square complexes. We deduce that many other properties of groups are undecidable. For hyperbolic groups, there cannot exist algorithms to determine largeness, the existence of a linear representation with infinite image (over any infinite field), or the rank of the profinite completion. [ABSTRACT FROM AUTHOR]

We prove that every finitely presented group with positive first ℓ²-Betti number that virtually surjects onto Z is acylindrically hyperbolic. In particular, this implies acylindrical hyperbolicity of finitely presented residually finite groups with positive first ℓ²-Betti number as well as groups of deficiency at least 2. [ABSTRACT FROM AUTHOR]

We give an elementary criterion on a group G for the map Aut(G) → Out(G) to split virtually. This criterion applies to many residually finite CAT(0) groups and hyperbolic groups, and in particular to all finitely generated Coxeter groups. As a consequence the outer automorphism group of any finitely generated Coxeter group is residually finite and virtually torsion-free. [ABSTRACT FROM AUTHOR]

We consider symmetries of spatial graphs in compact 3-manifolds described by smooth finite group actions. This paper provides a method for constructing an infinite family of hyperbolic spatial graphs with given symmetry by connecting spatial graph versions of hyperbolic tangles in 3-cells of polyhedral cell decompositions induced from triangulations of the 3-manifolds. This method is applicable also to the case of ideal triangulations. [ABSTRACT FROM AUTHOR]

Following the previous work of Nikulin and Agol, Belolipetsky, Storm, and Whyte, it is known that there exist only finitely many (totally real) number fields that can serve as fields of definition of arithmetic hyperbolic reflection groups. We prove a new bound on the degree nk of these fields in dimension 3: nk does not exceed 9. Combined with previous results of Maclachlan and Nikulin, this leads to a new bound nk≤25 which is valid for all dimensions. We also obtain upper bounds for the discriminants of these fields and give some heuristic results which may be useful for the classification of arithmetic hyperbolic reflection groups. [ABSTRACT FROM AUTHOR]

The Fibonacci groups F (n) are known to exhibit significantly different behaviour depending on the parity of n. We extend known results for F (n) for odd n to the family of Fractional Fibonacci groups F k / l (n). We show that for odd n the group F k / l (n) is not the fundamental group of an orientable hyperbolic 3-orbifold of finite volume. We obtain results concerning the existence of torsion in the groups F k / l (n) (where n is odd) paying particular attention to the groups F k (n) and F k / l (3) , and observe consequences concerning the asphericity of relative presentations of their shift extensions. We show that if F k (n) (where n is odd) and F k / l (3) are non-cyclic 3-manifold groups then they are isomorphic to the direct product of the quaternion group Q 8 and a finite cyclic group. [ABSTRACT FROM AUTHOR]

Let $\mathcal{C}$ be a variety of finite groups. We use profinite Bass--Serre theory to show that if u : H ↪ G is a map of finitely generated residually $\mathcal{C}$ groups such that the induced map û : Ĥ → Ĝ is a surjection of the pro-$\mathcal{C}$ completions, and G has more than one end, then H has the same number of ends as G. However if G has one end the number of ends of H may be larger; we observe cases where this occurs for $\mathcal{C}$ the class of finite p-groups.We produce a monomorphism of groups u : H ↪ G such that: either G is hyperbolic but not residually finite; or û : Ĥ → Ĝ is an isomorphism of profinite completions but H has property (T) (and hence (FA)), but G has neither. Either possibility would give new examples of pathological finitely generated groups. [ABSTRACT FROM PUBLISHER]

Let N be a finitely generated normal subgroup of a Gromov hyperbolic group . We establish criteria for N to have solvable conjugacy problem and be conjugacy separable in terms of the corresponding properties of . We show that the hyperbolic group from F. Haglund's and D. Wise's version of Rips's construction is hereditarily conjugacy separable. We then use this construction to produce first examples of finitely generated and finitely presented conjugacy separable groups that contain non-(conjugacy separable) subgroups of finite index. [ABSTRACT FROM AUTHOR]

In this paper we develop some of the foundations of the theory of relatively hyperbolic groups as originally formulated by Gromov. We prove the equivalence of two definitions of this notion. One is essentially that of a group admitting a properly discontinuous geometrically finite action on a proper hyperbolic space, that is, such that every limit point is either a conical limit point or a bounded parabolic point. The other is that of a group which admits a cofinite action on a connected fine hyperbolic graph. We define a graph to be "fine" if there are only finitely many circuits a given length containing any given edge, and we develop some of the properties of this notion. We show how a relatively hyperbolic group can be assumed to act on a proper hyperbolic space of a particular geometric form. We define the boundary of a relatively hyperbolic group, and show that the limit set of any geometrically finite action of the group is equivariantly homeomorphic to this boundary. This generalizes a result of Tukia for geometrically finite kleinian groups. We also describe when the boundary is connected. [ABSTRACT FROM AUTHOR]