3D hyperbolic Coxeter groups' growth rates are Perron numbers.
problem Growth rates of 3D hyperbolic Coxeter groups.
method Combining Parry's result and \cite{Y}'s main result.
result Growth rates of 3D hyperbolic Coxeter groups are Perron numbers.
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48 results for “3D hyperbolic Coxeter groups”
New proof shows not all Salem numbers are growth rates of Coxeter groups.
problem Identifying growth rates of Coxeter groups using Salem numbers.
method New proof using spectral radii and Coxeter transformations.
result Not every Salem number is a growth rate of hyperbolic Coxeter groups.
New groups found in hyperbolic 4D and 5D space have minimal growth rate.
problem Finding minimal growth rates in hyperbolic Coxeter groups.
method Combinatorial properties of hyperbolic Coxeter polyhedra, partial classification results, monotonicity properties of growth rates.
result Coxeter groups and in and have the smallest growth rate.
Random Coxeter Groupsmath.GR
Study on random Coxeter groups, extending Erdös-Rényi model.
problem Understanding random Coxeter groups and their properties.
method Extended Erdös-Rényi model to study random general Coxeter groups.
result Results on homology of nerve and hyperbolicity of random Coxeter groups.
For right-angled Coxeter groups , we obtain a condition on that is necessary and sufficient to ensure that is thick and thus not relatively hyperbolic. We show that Coxeter groups which are not thick all admit canonical minimal relatively hyperbolic structures; further, we show that in such a structure, …
Characterizes Coxeter groups with specific boundary shapes.
problem Identifying Coxeter groups with Sierpiński or Menger curve boundaries.
method Combining results from the literature on Gromov boundaries and Coxeter groups.
result Complete characterizations of hyperbolic Coxeter groups with Sierpiński or Menger curve boundaries.
Study growth rates of specific hyperbolic Coxeter groups with fixed dihedral angles.
problem Arithmetic properties of growth rates in hyperbolic Coxeter groups.
method Analyzing arithmetic properties of growth rates for specific Coxeter groups with fixed dihedral angles.
result Growth rates are always Perron numbers.
Threshold found for hyperbolicity in random Coxeter groups.
problem Determining the hyperbolicity threshold in random Coxeter groups.
method Analyzing random right-angled Coxeter groups via Erdős-Rényi graphs and combinatorial properties.
result Threshold for relative hyperbolicity in random Coxeter groups.
Proves minimal growth rate for Coxeter groups in hyperbolic space.
problem Finding minimal growth rate Coxeter groups in hyperbolic space.
method Generalizes methods from cocompact case to non-cocompact groups.
result Proves minimal growth rate for Coxeter groups .
We study relatively hyperbolic Coxeter groups of type with maximal Euclidean Coxeter subgroups of codimension 1. Our main result in this paper is that the dimension of these groups is bounded above.
Geometric constraints help classify hyperbolic polytopes.
problem Classifying reflective anisotropic Lorentzian lattices and cocompact arithmetic hyperbolic reflection groups.
method Established geometric constraints on compact Coxeter polytopes in hyperbolic spaces.
result Geometric constraints are useful for classifying hyperbolic polytopes.
We study homomorphisms from Kähler groups to Coxeter groups. As an application, we prove that a cocompact complex hyperbolic lattice (in complex dimension at least 2) does not embedd into a Coxeter group or a right-angled Artin group. This is in contrast with the case of real hyperbolic lattices.
Minimal covolume group found in hyperbolic 3-space.
problem Finding groups with minimal covolume in hyperbolic 3-space.
method Proved existence of a specific group with minimal covolume.
result Minimal covolume group found with covolume equal to Catalan's constant.
Proves involutions on Right-angled Coxeter groups without fixed points.
problem Fixed-point-free involutions on group boundaries.
method Analyzes Right-angled Coxeter groups, proving conjecture variation.
result Proves involutions without fixed points on boundaries of Right-angled Coxeter groups.
Paper constructs hyperbolic Coxeter groups that virtually fiber over Z.
problem Creating hyperbolic Coxeter groups that virtually fiber.
method Iterative procedure combining Jankiewicz-Norin-Wise results and Osajda's construction.
result Proves construction of groups with increasing vcd with each iteration.
We study combinatorial modulus on boundaries of hyperbolic Coxeter groups. We give new examples of hyperbolic groups whose boundary satisfies a combinatorial version of the Loewner property, and prove Cannon's conjecture for Coxeter groups. We also establish some connections with l^p cohomology.
Characterizes Coxeter groups with convex cocompact representations in projective space.
problem Understanding representations of Coxeter groups as convex cocompact reflection groups.
method Investigates representations of Coxeter groups into GL(n,R) as geometric reflection groups in projective space.
result Characterizes Coxeter groups that admit convex cocompact representations and describes the spaces of such representations.
New families of hyperbolic polyhedra yield infinitely many unique reflection groups.
problem Understanding commensurability classes of compact Coxeter polyhedra in hyperbolic spaces.
method Analyzing families of compact Coxeter polyhedra constructed by Makarov.
result Proves infinitely many commensurability classes in 4- and 5-dimensional hyperbolic spaces.
We give a necessary and sufficient condition for a hyperbolic Coxeter group with planar nerve to have Sierpiński curve as its Gromov boundary.
The boundary of certain hyperbolic groups is like a Menger curve.
problem Characterizing boundaries of hyperbolic Coxeter groups.
method Analyzing the nerve of hyperbolic right-angled Coxeter groups.
result Many triangulations and disks have boundaries homeomorphic to the Menger curve.
Study character varieties of a Coxeter group in hyperbolic and Anti-de Sitter spaces.
problem Characterize the geometric transitions of a Coxeter group's holonomy representations.
method Analysis of rigidity properties and character varieties in hyperbolic and Anti-de Sitter spaces.
result Description of singularity at the collapse of a right-angled cuboctahedron.
New groups found in anti-de Sitter space that are not quasi-isometric to hyperbolic space.
problem Finding strictly GHC-regular groups in anti-de Sitter space that are not quasi-isometric to hyperbolic space.
method Using the Tits representation of well-chosen Coxeter groups.
result Examples of strictly GHC-regular groups in anti-de Sitter space that are not quasi-isometric to hyperbolic space.
Cube complexes allow hyperbolic groups to have Anosov representations.
problem Understanding representations of hyperbolic groups in higher dimensions.
method Analyzing representations of hyperbolic groups acting on CAT(0) cube complexes.
result Generic representations of certain groups are Anosov.
We study the set G of growth rates of of ideal Coxeter groups in hyperbolic 3-space which consists of real algebraic integers greater than 1. We show that (1) G is unbounded above while it has the minimum, (2) any element of G is a Perron number, and (3) growth rates of of ideal Coxeter groups with generators are l…
New groups algebraically fibre with high-dimensional hyperbolic groups.
problem Finding new quasi-isometry classes of hyperbolic groups.
method Constructing infinitely many hyperbolic groups as finite-index subgroups of right-angled Coxeter groups.
result Groups algebraically fibre with finitely presented kernels, expanding finiteness properties.
New groups with Menger curve boundaries found.
problem Finding non-hyperbolic groups with specific boundary shapes.
method Using Coxeter groups with complete graph nerves and embedding theorems.
result First non-hyperbolic groups with Menger curve boundaries discovered.
In this paper, we compute the covolume of the group of units of the quadratic form f_d^n(x) = x_1^2 + x_2^2 + . . . + x_n^2 - d x_{n+1}^2 with d an odd, positive, square-free integer. Mcleod has determined the hyperbolic Coxeter fundamental domain of the reflection subgroup of the group of units of the quadratic form f…
Commensurability for certain right-angled Coxeter groups and geometric amalgams of free groupsmath.GR
Conditions for commensurability in specific groups are derived.
problem Determining when certain groups are commensurable.
method New geometric realization of right-angled Coxeter groups and surface amalgams.
result Explicit conditions for commensurability of specific groups.
The reflection length of an element of a Coxeter group is the minimal number of conjugates of the standard generators whose product is equal to that element. In this paper we prove the conjecture of McCammond and Petersen that reflection length is unbounded in any non-affine Coxeter group. Among the tools used, the con…
Abstract Coxeter groups have growth rates that are Perron numbers.
problem Understanding growth rates of Coxeter groups.
method Defined a class of Coxeter groups, --spanned, and analyzed their growth rates.
result For --spanned Coxeter groups, geodesic growth rate strictly dominates word growth rate and appears to be a Perron number.
Small sets of systoles fill hyperbolic surfaces of large genus.
problem Constructing hyperbolic surfaces with minimal systole sets.
method Theory of Coxeter groups combined with number theory.
result Cardinality of systole sets is in o(g/ ln g) for large genus.
Constructs proper affine actions for right-angled Coxeter groups.
problem Proper affine actions for right-angled Coxeter groups.
method Constructs proper actions of right-angled Coxeter groups on O(p,q+1) and its Lie algebra by affine transformations.
result Any virtually special group admits proper affine actions on some R^n.
In [6], Kellerhals and Perren conjectured that the growth rates of the reflection groups given by hyperbolic Coxeter polyhedra are always Perron numbers. We prove that this conjecture is always true for the case of ideal Coxeter polyhedra in . We also find out the ideal Coxeter polyhedron in $\mathbb{H}^3…
The rich theory of Coxeter groups is used to provide an algebraic construction of finite volume hyperbolic n-manifolds. Combinatorial properties of finite images of these groups can be used to compute the volumes of the resulting manifolds. Three examples, in 4,5 and 6-dimensions, are given, each of very small volume, …
New method to study group invariants using divergence spectra.
problem Understanding group invariants through divergence.
method Introducing divergence spectrum to compare classical notions and study relatively hyperbolic groups.
result Existence of groups with exponential divergence but different divergence spectra.
Study on right-angled Coxeter groups and their geometric properties.
problem Characterizing the coarse geometry of right-angled Coxeter groups.
method Analyzing graph properties and applying geometric group theory.
result Proves properties of right-angled Coxeter groups, including quasi-isometry and divergence.
Bounds on conformal dimension for certain Coxeter group boundaries.
problem Determining conformal dimensions for specific Coxeter group boundaries.
method Quasi-isometric embeddings and geometric actions on CAT(-1) spaces.
result Upper and lower bounds on conformal dimension for specified Coxeter groups.
The study examines connections between Coxeter groups and their alternating quotients.
problem Characterizing connections between Coxeter groups and their alternating quotients.
method Analyzes the structure of right-angled Coxeter groups and their quasiconvex subgroups.
result Establishes conditions for the connectivity of alternating quotients of Coxeter groups.
Characterizes strongly quasiconvex subsets in hierarchically hyperbolic spaces.
problem Understanding the structure of strongly quasiconvex subsets in HHSs.
method Characterization through contracting properties, relative divergence, and hierarchical structure.
result Proves characterization of hyperbolically embedded subgroups in hierarchically hyperbolic groups.
Bowditch's JSJ tree for splittings over 2-ended subgroups is a quasi-isometry invariant for 1-ended hyperbolic groups which are not cocompact Fuchsian. Our main result gives an explicit, computable "visual" construction of this tree for certain hyperbolic right-angled Coxeter groups. As an application of our constructi…
For a finite volume geodesic polyhedron P in hyperbolic 3-space, with the property that all interior angles between incident faces are integral submultiples of Pi, there is a naturally associated Coxeter group generated by reflections in the faces. Furthermore, this Coxeter group is a lattice inside the isometry group …
New method constructs non-quasiconvex subgroups in hyperbolic groups.
problem Creating non-quasiconvex subgroups in hyperbolic groups.
method Using Stallings-like techniques on right-angled Coxeter groups (RACGs).
result Explicit examples of non-quasiconvex subgroups constructed.
Study complex reflections in infinite Coxeter tetrahedron moduli space.
problem Characterize representations of Coxeter group in complex hyperbolic space.
method Type-preserving representations of Coxeter group to , parameterized by .
result Discrete and faithful representations for . First nontrivial moduli space in complex hyperbolic space.
We prove the following: there are infinitely many finite-covolume (resp. cocompact) Coxeter groups acting on hyperbolic space H^n for every n < 20 (resp. n < 7). When n=7 or 8, they may be taken to be nonarithmetic. Furthermore, for 1 < n < 20, with the possible exceptions n=16 and 17, the number of essentially distinc…
New methods classify hyperbolic polytopes with up to 40 facets.
problem Classifying compact hyperbolic Coxeter polytopes with specific facet counts.
method New combinatorial method via point set order types.
result Proves existence of a compact hyperbolic Coxeter 29-polytope with at least 40 facets.
The notion of limit roots of a Coxeter group W was recently introduced (see arXiv:1112.5415 and arXiv:1303.6710): they are the accumulation points of directions of roots of a root system for W. In the case where the root system lives in a Lorentzian space W admits a faithful representation as a discrete reflection grou…
New insights into Coxeter groups' -cohomology via weighted analysis.
problem Understanding weighted -cohomology of Coxeter groups.
method Using weighted -cohomology groups and complexes called ruins, along with the Strong Atiyah Conjecture.
result Weighted -cohomology groups of Coxeter groups are concentrated in low dimensions.
Minimal non-arithmetic hyperbolic 3-orbifold found with least volume.
problem Finding the hyperbolic 3-orbifold with minimal volume among non-arithmetic ones.
method Utilized the tetrahedral Coxeter group and horoball configuration to prove minimal volume.
result The 1-cusped quotient of hyperbolic space by the tetrahedral Coxeter group has minimal volume.