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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]

In a discrete group generated by hyperplane reflections in the 푛-dimensional hyperbolic space, the reflection length of an element is the minimal number of hyperplane reflections in the group that suffices to factor the element. For a Coxeter group that arises in this way and does not split into a direct product of spherical and affine reflection groups, the reflection length is unbounded. The action of the Coxeter group induces a tessellation of the hyperbolic space. After fixing a fundamental domain, there exists a bijection between the tiles and the group elements. We describe certain points in the visual boundary of the 푛-dimensional hyperbolic space for which every neighbourhood contains tiles of every reflection length. To prove this, we show that two disjoint hyperplanes in the 푛-dimensional hyperbolic space without common boundary points have a unique common perpendicular. [ABSTRACT FROM AUTHOR]

Let $\Sigma $ be a nonempty subset of the set of prime numbers which is either equal to the entire set of prime numbers or of cardinality one. In the present paper, we continue our study of the pro- $\Sigma $ fundamental groups of hyperbolic curves and their associated configuration spaces over algebraically closed fields in which the primes of $\Sigma $ are invertible. The present paper focuses on the topic of comparison between the theory developed in earlier papers concerning pro- $\Sigma $ fundamental groups and various discrete versions of this theory. We begin by developing a theory concerning certain combinatorial analogues of the section conjecture and Grothendieck conjecture. This portion of the theory is purely combinatorial and essentially follows from a result concerning the existence of fixed points of actions of finite groups on finite graphs (satisfying certain conditions). We then examine various applications of this purely combinatorial theory to scheme theory. Next, we verify various results in the theory of discrete fundamental groups of hyperbolic topological surfaces to the effect that various properties of (discrete) subgroups of such groups hold if and only if analogous properties hold for the closures of these subgroups in the profinite completions of the discrete fundamental groups under consideration. These results make possible a fairly straightforward translation , into discrete versions , of pro- $\Sigma $ results obtained in previous papers by the authors. Finally, we discuss a construction that was considered previously by M. Boggi in the discrete case from the point of view of the present paper. [ABSTRACT FROM AUTHOR]

We prove a finiteness theorem for subgroups of bounded rank in hyperbolic 3‐manifold groups. As a consequence, we show that every bounded rank covering tower of closed hyperbolic 3‐manifolds is a tower of finite covers associated to a fibration over a 1‐orbifold. [ABSTRACT FROM AUTHOR]

We show the existence of a weak solution for a system of partial differential equations describing the motion of a flexible solid inside a fluid: A nonlinear, viscoelastic, n-dimensional bulk solid governed by a PDE including inertia is interacting with an incompressible fluid governed by the (n-dimensional) Navier-Stokes equation for n ≥ 2. The result is the first allowing for large bulk deformations in the regime of long time existence for fluid-structure interactions. The existence is achieved by introducing a novel variational scheme involving two time-scales that allows us to extend the method of minimizing movements to hyperbolic problems involving nonconvex and degenerate energies. [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]

We study Frattini subgroups of various generalizations of hyperbolic groups. For any countable group G admitting a general type action on a hyperbolic space S , we show that the induced action of the Frattini subgroup Φ (G) on S has bounded orbits. This implies that Φ (G) is "small" compared to G ; in particular, | G : Φ (G) | = ∞. In contrast, for any finitely generated non-cyclic group Q with Φ (Q) = { 1 } , we construct an infinite lacunary hyperbolic group L such that L / Φ (L) ≅ Q ; in particular, the Frattini subgroup of an infinite lacunary hyperbolic group can have finite index. As an application, we obtain the first examples of invariably generated, infinite, lacunary hyperbolic groups. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the two-weight Hardy inequalities on metric measure space possessing polar decompositions for the case p = 1 and 1 ≤ q < ∞ . This result complements the Hardy inequalities obtained in [M. Ruzhansky and D. Verma, Hardy inequalities on metric measure spaces, Proc. Roy. Soc. A. 475 2019, 2223, Article ID 20180310] in the case 1 < p ≤ q < ∞ . The case p = 1 requires a different argument and does not follow as the limit of known inequalities for p > 1 . As a byproduct, we also obtain the best constant in the established inequality. We give examples obtaining new weighted Hardy inequalities on homogeneous Lie groups, on hyperbolic spaces and on Cartan–Hadamard manifolds for the case p = 1 and 1 ≤ q < ∞ . [ABSTRACT FROM AUTHOR]

We show that two-dimensional Artin groups satisfy a strengthening of the Tits alternative: their subgroups either contain a non-abelian free group or are virtually free abelian of rank at most 2. When in addition the associated Coxeter group is hyperbolic, we answer in the affirmative a question of Wise on the subgroups generated by large powers of two elements: given any two elements a , b of a two-dimensional Artin group of hyperbolic type, there exists an integer n ≥ 1 such that a n and b n either commute or generate a non-abelian free subgroup. [ABSTRACT FROM AUTHOR]

Let $\Gamma =\langle I_{1}, I_{2}, I_{3}\rangle $ be the complex hyperbolic $(4,4,\infty)$ triangle group with $I_1I_3I_2I_3$ being unipotent. We show that the limit set of $\Gamma $ is connected and the closure of a countable union of $\mathbb {R}$ -circles. [ABSTRACT FROM AUTHOR]

Let G be a representation-theoretic Kac–Moody group associated to a nonsingular symmetrizable generalized Cartan matrix. We first consider Kac–Moody analogs of Borel Eisenstein series (induced from quasicharacters on the Borel), and prove they converge almost everywhere inside the Tits cone for arbitrary spectral parameters in the Godement range. We then use this result to show the full absolute convergence everywhere inside the Tits cone (again for spectral parameters in the Godement range) for a class of Kac–Moody groups satisfying a certain combinatorial property, in particular for rank-2 hyperbolic groups. [ABSTRACT FROM AUTHOR]

In this paper first, the boundedness and compactness of a composition operator from Zygmund space to Besov Zygmund-type space are studied. Then we study this concepts for this operator by using the hyperbolic-type analytic Besov Zygmund-type class. Finally, we show the relation between the hyperbolictype analytic Besov Zygmund-type class and the meromorphic (or spherical) Besov Zygmund-type class. [ABSTRACT FROM AUTHOR]

In this paper, we consider convex co-compact subgroups of the projective linear group. We prove that such a group is relatively hyperbolic with respect to a collection of virtually Abelian subgroups of rank 2 if and only if each open face in the ideal boundary has dimension at most one. We also introduce the “coarse Hilbert dimension” of a subset of a convex set and use it to characterize when a naive convex co-compact subgroup is word hyperbolic or relatively hyperbolic with respect to a collection of virtually Abelian subgroups of rank 2. [ABSTRACT FROM AUTHOR]

We prove that, for any hyperbolic group, the compressed word and the compressed conjugacy problems are solvable in polynomial time. As a consequence, the word problem for the (outer) automorphism group of a hyperbolic group is solvable in polynomial time. We also prove that the compressed simultaneous conjugacy and the compressed centraliser problems are solvable in polynomial time. Finally, we prove that, for any infinite hyperbolic group, the compressed knapsack problem is NP-complete. [ABSTRACT FROM AUTHOR]

Given a group G acting by isometries on a metric space X, we consider a preferred collection of paths of the space X, a path system, and study the spectrum of relative exponential growth rates and quotient exponential growth rates of the infinite index subgroups of G that are quasi-convex with respect to this path system. If G contains a constricting element with respect to the same path system, we are able to determine when the growth rates of the first kind are strictly smaller than the growth rate of G, and when the growth rates of the second kind coincide with the growth rate of G. Examples of applications include relatively hyperbolic groups, CAT.0/ groups, and hierarchically hyperbolic groups containing a Morse element. [ABSTRACT FROM AUTHOR]

We produce an example demonstrating that every finitely generated relatively hyperbolic group with respect to a collection of Hopfian subgroups need not be Hopfian. This answers a question of Osin (2006) in the negative. [ABSTRACT FROM AUTHOR]

In this paper, we define a set of elementary orthogonal matrices similar to Vaserstein's definition of elementary symplectic matrices. We prove that the elementary orthogonal group generated by these matrices is a conjugate of the Dickson–Siegel–Eichler–Roy (DSER) elementary orthogonal group defined on a free quadratic module with a hyperbolic summand. [ABSTRACT FROM AUTHOR]

We give a simple combinatorial criterion, in terms of an action on a hyperbolic simplicial complex, for a group to be hierarchically hyperbolic. We apply this to show that quotients of mapping class groups by large powers of Dehn twists are hierarchically hyperbolic (and even relatively hyperbolic in the genus 2 case). In genus at least three, there are no known infinite hyperbolic quotients of mapping class groups. However, using the hierarchically hyperbolic quotients we construct, we show, under a residual finiteness assumption, that mapping class groups have many nonelementary hyperbolic quotients. Using these quotients, we relate questions of Reid and Bridson–Reid–Wilton about finite quotients of mapping class groups to residual finiteness of specific hyperbolic groups. [ABSTRACT FROM AUTHOR]

We consider the initial problem for one class of fractional stochastic differential equations with white noise in time and space for solutions of distribution type. Existence and singularity theorems for solutions are established and explicit representations for them are found. [ABSTRACT FROM AUTHOR]

We classify in terms of Hopf-type properties mapping tori of residually finite Poincaré Duality groups with non-zero Euler characteristic. This generalises and gives a new proof of the analogous classification for fibered 3-manifolds. Various applications are given. In particular, we deduce that rigidity results for Gromov hyperbolic groups hold for the above mapping tori with trivial center. [ABSTRACT FROM AUTHOR]

We show that any non-standard generating pair of a hyperbolic triangle group is represented by a special almost orbifold covering with a good marking. [ABSTRACT FROM AUTHOR]

Let G 6 , 3 be a hyperbolic polygon-group with boundary the Menger curve. Granier (Groupes discrets en géométrie hyperbolique—aspects effectifs, Université de Fribourg, 2015) constructed a discrete, convex cocompact and faithful representation ρ of G 6 , 3 into PU (2 , 1) . We show the 3-orbifold at infinity of ρ (G 6 , 3) is a closed hyperbolic 3-orbifold, with underlying space the 3-sphere and singular locus the Z 3 -coned chain-link C (6 , - 2) . This answers the second part of Kapovich’s Conjecture 10.6 in Kapovich (in: In the tradition of thurston II. Geometry and groups, Springer, Cham, 2022), and it also provides the second explicit example of a closed hyperbolic 3-orbifold that admits a uniformizable spherical CR-structure after Schwartz’s first example in Schwartz (Invent Math 151(2):221–295, 2003). [ABSTRACT FROM AUTHOR]

We prove that the compressed conjugacy problem in a group that is hyperbolic relative to a collection of free abelian subgroups is solvable in polynomial time. [ABSTRACT FROM AUTHOR]

Bestvina's notion of a Z-structure provides a general framework for group boundaries that includes Gromov boundaries of hyperbolic groups and visual boundaries of CAT(0) groups as special cases. A refinement, known as an EZ-structure, has proven useful in attacks on the Novikov conjecture and related problems. Characterizations of groups admitting a Z- or an EZ-structure are longstanding open problems. In this paper, we examine groups of the form G ... Z. For example, we show that if G is torsion-free and admits a Z-structure, then so does every semidirect product of this type. We prove a similar theorem for EZ-structures, under an additional hypothesis. As applications, we show that all closed 3-manifold groups admit Z-structures, as do all strongly polycyclic groups and all groups of polynomial growth. In those latter cases, our Z-boundaries are always spheres. This allows one to make strong conclusions about the group cohomology and end invariants of those groups. In another direction, we expand upon the notion of an EZ-structure and discuss new applications to the Novikov conjecture. [ABSTRACT FROM AUTHOR]

Let Γ be a Fuchsian group acting on the hyperbolic upper half-plane H , such that Γ \ H is a geometrically finite Riemann surface with respect to the natural hyperbolic metric induced from H. If γ is hyperbolic then following [J. Kudla and S. Millson, Harmonic differentials and closed geodesics on a Riemann surface, Invent. Math. 54 (1979) 193–211; J. D. Fay, Fourier coefficients of the resolvent for a Fuchsian group, J. Reine Angew. Math. 293 (1977) 143–203], there is a corresponding hyperbolic Eisenstein series. In this paper, we study the limiting behavior of hyperbolic Eisenstein series on a degenerating family of geometrically finite hyperbolic surfaces. In particular, we give a partial lightening to a question of Ji, concerning the approximation of Eisenstein series during degeneration (see Proposition 5.2 and Theorem 5.2). [ABSTRACT FROM AUTHOR]

Let (Γ,P)$(\Gamma,\mathbb {P})$ be a relatively hyperbolic group pair that is relatively one ended. Then, the Bowditch boundary of (Γ,P)$(\Gamma,\mathbb {P})$ is locally connected. Bowditch previously established this conclusion under the additional assumption that all peripheral subgroups are finitely presented, either one or two ended, and contain no infinite torsion subgroups. We remove these restrictions; we make no restriction on the cardinality of Γ$\Gamma$ and no restriction on the peripheral subgroups P∈P$P \in \mathbb {P}$. [ABSTRACT FROM AUTHOR]

We prove that the Gromov hyperbolic groups obtained by the strict hyperbolization procedure of Charney and Davis are virtually compact special, hence linear and residually finite. Our strategy consists in constructing an action of a hyperbolized group on a certain dual CAT (0) cubical complex. As a result, all the common applications of strict hyperbolization are shown to provide manifolds with virtually compact special fundamental group. In particular, we obtain examples of closed negatively curved Riemannian manifolds whose fundamental groups are linear and virtually algebraically fiber. [ABSTRACT FROM AUTHOR]

Actions on hyperbolic metric spaces are an important tool for studying groups, and so it is natural, but difficult, to attempt to classify all such actions of a fixed group. In this paper, we build strong connections between hyperbolic geometry and commutative algebra in order to classify the cobounded hyperbolic actions of numerous metabelian groups up to a coarse equivalence. In particular, we turn this classification problem into the problems of classifying ideals in the completions of certain rings and calculating invariant subspaces of matrices. We use this framework to classify the cobounded hyperbolic actions of many abelian‐by‐cyclic groups associated to expanding integer matrices. Each such action is equivalent to an action on a tree or on a Heintze group (a classically studied class of negatively curved Lie groups). Our investigations incorporate number systems, factorization in formal power series rings, completions, and valuations. [ABSTRACT FROM AUTHOR]

Given a non‐positively curved cube complex X$X$, we prove that the quotient of π1X$\pi _1X$ defined by a cubical presentation ⟨X∣Y1,⋯,Ys⟩$\langle X\mid Y_1,\dots, Y_s\rangle$ satisfying sufficient non‐metric cubical small‐cancellation conditions is hyperbolic provided that π1X$\pi _1X$ is hyperbolic. This generalises the fact that finitely presented classical C(7)$C(7)$ small‐cancellation groups are hyperbolic. [ABSTRACT FROM AUTHOR]

The Bulletin of Symbolic Logic contains various announcements and information relevant to the field of mathematical logic. It includes details about the 2024 Sacks Prize for outstanding doctoral dissertations, the Turing Award recipient, ASL membership renewal, and opportunities for student travel awards. The document also highlights upcoming logic conferences, ASL-sponsored meetings, and guidelines for abstract submissions. Additionally, it provides information on ASL membership options, discounts, and free membership programs for individuals in developing economies. [Extracted from the article]

Given the importance of hyperbolic quantum color codes and Euclidean quantum color codes, this paper considers the study of the former codes on compact surfaces with genus g ≥ 2 from the mathematical point of view. Identifying the normal subgroup in the decomposition of the full symmetry group of the { p , 3 } tessellation is relevant because it provides the algebraic structure for identifying and constructing a class of linear shrunk hyperbolic quantum color codes. Under this assumption, the normal subgroup's presentation, the whole process's kernel, is derived from the Reidemeister–Schreier method. As a result, we present a class of regular normal hyperbolic quantum color codes derived from the { 6 j , 3 } tessellation with encoding rate going asymptotically to 1. The regular tessellation { 6 j , 3 } includes the two types of tessellations: (1) the densest tessellation { 12 i - 6 , 3 } when j = 2 i - 1 and (2) the tessellation { 12 i , 3 } when j = 2 i , for i ∈ N . An analysis of the minimum distance achieved by this class of regular normal hyperbolic quantum color codes is performed. [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 work, we describe the moduli of triples of points in quaternionic projective space which define uniquely the congruence classes of such triples relative to the action of the isometry group of quaternionic hyperbolic space H Q n. To solve this problem, we introduce some basic invariants of triples of points in quaternionic hyperbolic geometry. In particular, we define quaternionic analogues of the Goldman invariants for mixed configurations of points introduced by him in complex hyperbolic geometry. [ABSTRACT FROM AUTHOR]

We investigate barycenters of probability measures on Gromov hyperbolic spaces, toward development of convex optimization in this class of metric spaces. We establish a contraction property (the Wasserstein distance between probability measures provides an upper bound of the distance between their barycenters), a deterministic approximation of barycenters of uniform distributions on finite points, and a kind of law of large numbers. These generalize the corresponding results on CAT(0)-spaces, up to additional terms depending on the hyperbolicity constant. [ABSTRACT FROM AUTHOR]

In this paper, we construct signal constellations from lattices in complex hyperbolic spaces. To construct a hyperbolic lattice, we identify an arithmetic Fuchsian group with the group of units O1 of a natural quaternion order O ⊂ V, in which V is some quaternion algebra over an algebraic number field. The arithmetic Fuchsian group, G, is isomorphic to the fundamental group of a regular hyperbolic polygon P, and an oriented compact surface arises from the pairwise identification of its opposite edges. The polygon P is the fundamental region associated with a regular tessellation {p, q}. The main contribution of this paper is to employ the Reidemeister-Schreier rewriting process to use proper decompositions of the full symmetry group of a tessellation {p, q}, which allows the matching of uniform signal constellations to quotient groups of G. In this direction, we consider the labelling of phaseshift keying (PSK) and amplitude-phase keying (APK) signal constellations diagrams via three approaches: cyclic quotient groups, a direct product of cyclic quotient groups and semi-direct product of cyclic groups (the dihedral group case). [ABSTRACT FROM AUTHOR]

Hyperbolic lattices are a new type of synthetic materials based on regular tessellations in non-Euclidean spaces with constant negative curvature. While so far, there has been several theoretical investigations of hyperbolic topological media, experimental work has been limited to time-reversal invariant systems made of coupled discrete resonances, leaving the more interesting case of robust, unidirectional edge wave transport completely unobserved. Here, we report a non-reciprocal hyperbolic network that exhibits both Chern and anomalous chiral edge modes, and implement it on a planar microwave platform. We experimentally evidence the unidirectional character of the topological edge modes by direct field mapping. We demonstrate the topological origin of these hyperbolic chiral edge modes by an explicit topological invariant measurement, performed from external probes. Our work extends the reach of topological wave physics by allowing for backscattering-immune transport in materials with synthetic non-Euclidean behavior. Here the authors experimentally demonstrate the anomalous and Chern topological phases in a hyperbolic non-reciprocal scattering network, establishing unidirectional channels to induce new and exciting wave transport properties in curved spaces. [ABSTRACT FROM AUTHOR]

Let |$\alpha $| and |$\beta $| be uniformly random permutations of orders |$2$| and |$3$|⁠ , respectively, in |$S_{N}$|⁠ , and consider, say, the permutation |$\alpha \beta \alpha \beta ^{-1}$|⁠. How many fixed points does this random permutation have on average? The current paper studies questions of this kind and relates them to surprising topological and algebraic invariants of elements in free products of groups. Formally, let |$\Gamma =G_{1}*\ldots *G_{k}$| be a free product of groups where each of |$G_{1},\ldots ,G_{k}$| is either finite, finitely generated free, or an orientable hyperbolic surface group. For a fixed element |$\gamma \in \Gamma $|⁠ , a |$\gamma $| -random permutation in the symmetric group |$S_{N}$| is the image of |$\gamma $| through a uniformly random homomorphism |$\Gamma \to S_{N}$|⁠. In this paper we study local statistics of |$\gamma $| -random permutations and their asymptotics as |$N$| grows. We first consider |$\mathbb{E}\big [\textrm{fix}_{\gamma }\big (N\big)\big ]$|⁠ , the expected number of fixed points in a |$\gamma $| -random permutation in |$S_{N}$|⁠. We show that unless |$\gamma $| has finite order, the limit of |$\mathbb{E}\big [\textrm{fix}_{\gamma }\big (N\big)\big ]$| as |$N\to \infty $| is an integer, and is equal to the number of subgroups |$H\le \Gamma $| containing |$\gamma $| such that |$H\cong \mathbb{Z}$| or |$H\cong C_{2}*C_{2}$|⁠. Equivalently, this is the number of subgroups |$H\le \Gamma $| containing |$\gamma $| and having (rational) Euler characteristic zero. We also prove there is an asymptotic expansion for |$\mathbb{E}\big [\textrm{fix}_{\gamma }\big (N\big)\big ]$| and determine the limit distribution of the number of fixed points as |$N\to \infty $|⁠. These results are then generalized to all statistics of cycles of fixed lengths. [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]

Consider a non-elementary Gromov-hyperbolic group Γ with a suitable invariant hyperbolic metric, and an ergodic probability measure preserving (p.m.p.) action on (X , μ) . We construct special increasing sequences of finite subsets F n (y) ⊂ Γ , with (Y , ν) a suitable probability space, with the following properties. Given any countable partition P of X of finite Shannon entropy, the refined partitions ⋁ γ ∈ F n (y) γ P have normalized information functions which converge to a constant limit, for μ -almost every x ∈ X and ν -almost every y ∈ Y . The sets F n (y) constitute almost-geodesic segments, and ⋃ n ∈ N F n (y) is a one-sided almost geodesic with limit point F + (y) ∈ ∂ Γ , starting at a fixed bounded distance from the identity, for almost every y ∈ Y . The distribution of the limit point F + (y) belongs to the Patterson–Sullivan measure class on ∂ Γ associated with the invariant hyperbolic metric. The main result of the present paper amounts therefore to a Shannon–McMillan–Breiman theorem along almost-geodesic segments in any p.m.p. action of Γ as above. For several important classes of examples we analyze, the construction of F n (y) is purely geometric and explicit. Furthermore, consider the infimum of the limits of the normalized information functions, taken over all Γ -generating partitions of X. Using an important inequality due to Seward (Weak containment and Rokhlin entropy, arxiv:1602.06680, 2016), we deduce that it is equal to the Rokhlin entropy h Rok of the Γ -action on (X , μ) defined in Seward (Invent Math 215:265–310, 2019), provided that the action is free. Remarkably, this property holds for every choice of invariant hyperbolic metric, every choice of suitable auxiliary space (Y , ν) and every choice of special family F n (y) as above. In particular, for every ϵ > 0 , there is a generating partition P ϵ , such that for almost every y ∈ Y , the partition refined using the sets F n (y) has most of its atoms of roughly constant measure, comparable to exp (- n h Rok ± ϵ) . This describes an approximation to the Rokhlin entropy in geometric and dynamical terms, for actions of word-hyperbolic groups. [ABSTRACT FROM AUTHOR]

We show that low-density random quotients of cubulated hyperbolic groups are again cubulated (and hyperbolic). Ingredients of the proof include cubical small-cancellation theory, the exponential growth of conjugacy classes, and the statement that hyperplane stabilizers grow exponentially more slowly than the ambient cubical group. [ABSTRACT FROM AUTHOR]

We study relations between maps between relatively hyperbolic groups/spaces and quasisymmetric embeddings between their boundaries. More specifically, we establish a correspondence between (not necessarily coarsely surjective) quasi-isometric embeddings between relatively hyperbolic groups/spaces that coarsely respect peripherals, and quasisymmetric embeddings between their boundaries satisfying suitable conditions. Further, we establish a similar correspondence regarding maps with at most polynomial distortion. We use this to characterise groups which are hyperbolic relative to some collection of virtually nilpotent subgroups as exactly those groups which admit an embedding into a truncated real hyperbolic space with at most polynomial distortion, generalising a result of Bonk and Schramm for hyperbolic groups. [ABSTRACT FROM AUTHOR]

We prove that a word hyperbolic group whose Gromov boundary properly contains a 2‐sphere cannot admit a projective Anosov representation into Sp2m(C)$\mathsf {Sp}_{2m}(\mathbb {C})$, m∈N$m\in \mathbb {N}$. We also prove that a word hyperbolic group that admits a projective Anosov representation into Sp2m(R)$\mathsf {Sp}_{2m}(\mathbb {R})$ is virtually a free group or virtually a surface group, a result established independently by Dey–Greenberg–Riestenberg. [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]

We prove that in a cocompact complex hyperbolic arithmetic lattice Γ < PU (m , 1) of the simplest type, deep enough finite index subgroups admit plenty of homomorphisms to ℤ with kernel of type F m − 1 but not of type 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 Kähler manifolds. [ABSTRACT FROM AUTHOR]

Rips and Sela (1995) introduced the notion of globally stable cylinders and asked if all Gromov hyperbolic groups admit such. We prove that hyperbolic cubulated groups admit globally stable cylinders. [ABSTRACT FROM AUTHOR]

Hyperbolic triangle groups are found within the category of finitely generated groups. These are topological groups formed by the reflections along the sides of a hyperbolic triangle and acting properly discontinuously on the hyperbolic plane. Higman raised a question about the simplicity of finitely generated groups. The best known example of a simple group is the alternating group A n , where n ≥ 5 . This article establishes a relation between the hyperbolic triangle group denoted as ▵ * (3 , 7 , r) and the alternating group. The approach involves employing coset diagrams to establish this connection. The construction of adjacency matrices for these coset diagrams is performed, followed by a detailed examination of their spectral characteristics. [ABSTRACT FROM AUTHOR]

Given a simple connected compact Lie group K and a maximal torus T of K , the Weyl group W = N K (T) / T naturally acts on T. First, we use the combinatorics of the (extended) affine Weyl group to provide an explicit W -equivariant triangulation of T. We describe the associated W -dg-ring. For a non-crystallographic Coxeter group W , using compact hyperbolic extensions rather than affine ones, we construct a compact W -manifold T (W) , which is an analogue of a torus for W. We exhibit a W -equivariant triangulation of T (W) and compute the associated W -dg-ring. Also, we derive its homology representation. [ABSTRACT FROM AUTHOR]