Study groups with polynomial growth, finding structure and applications.
problem Understanding groups with polynomial growth structure.
method Structure theorem for locally compact groups of polynomial growth.
result Applications on various growth functions and relations to FC-G series.
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.
Study confined subgroups in groups with contracting elements, showing their growth rate is strictly greater than half of the ambient growth rate.
problem Understanding the growth rate of confined subgroups in groups with contracting elements.
method Through boundary actions, analyzing the Hopf decomposition and quotient growth.
result Confined subgroups have a growth rate strictly greater than half of the ambient growth rate.
Study growth rates of subgroups in groups with a constricting element.
problem Understanding growth rates of subgroups in groups with a constricting element.
method Examining the spectrum of relative and quotient exponential growth rates of quasi-convex subgroups.
result Determine when growth rates of subgroups are strictly smaller or coincide with the group's growth rate.
Groups on CAT(0) cube complexes grow exponentially uniformly.
problem Uniform exponential growth of groups acting on CAT(0) cube complexes.
method Study groups acting without global fixed points on CAT(0) square complexes.
result Groups with uniform exponential growth or stabilize Euclidean subcomplexes.
Homology growth of specific mapping tori vanishes for certain groups.
problem Homology growth of polynomially growing mapping tori in various groups.
method Proof of the cheap rebuilding property for specific groups.
result Torsion homology growth vanishes for Farber sequences in every degree.
Extends growth properties of hyperbolic groups to their extensions.
problem Quantifying subgroup alternatives in group laws.
method Develops a framework for preserving exponential growth in extensions of hyperbolic groups.
result Automorphism groups of certain hyperbolic and Artin groups have locally uniform exponential growth.
Compact groups with polynomial growth have specific embeddings.
problem Understanding groups with polynomial growth.
method Embedding into semidirect products of Lie and compact groups.
result Groups can be embedded as co-compact subgroups.
The study examines how automorphism growth rates of a group can be deduced from its simpler decompositions.
problem Determine automorphism growth rates of a group from its simpler decompositions.
method Analyze group decompositions into simpler pieces (direct products, free products, graph of groups) and deduce growth rates.
result Information about automorphism growth rates of a group can be deduced from its simpler decompositions.
Sharp growth tightness proven for group quotients.
problem Growth behavior of group quotients by confined subgroups.
method Statistically convex-cocompact action with contracting elements.
result Sharp growth tightness proven, with applications to uniformly recurrent subgroups.
New proofs for growth series of Coxeter groups using complex structures.
problem Proving new formulae for growth series of Coxeter groups.
method Using the structure of Coxeter complexes, Davis complexes, or Tits non-complexes.
result Several classical formulae for growth series are proved in a new way.
We give several sufficient conditions for uniform exponential growth in the setting of virtually torsion-free hierarchically hyperbolic groups. For example, any hierarchically hyperbolic group that is also acylindrically hyperbolic has uniform exponential growth. In addition, we provide a quasi-isometric characterizati…
We introduce and systematically study the concept of a growth tight action. This generalizes growth tightness for word metrics as initiated by Grigorchuk and de la Harpe. Given a finitely generated, non-elementary group G acting on a G--space X, we prove that if G contains a strongly contracting eleme…
We define a large class of abstract Coxeter groups, that we call ∞--spanned, and for which the word growth rate and the geodesic growth rate appear to be Perron numbers. This class contains a fair amount of Coxeter groups acting on hyperbolic spaces, thus corroborating a conjecture by Kellerhals and Perren. We a…
Study on volume growth of horospheres in specific Heintze groups.
problem Volume growth of horospheres in diagonalizable Heintze groups.
method Analysis of volume growth in Heintze groups with diagonalizable structure.
result Two distinct isometry and quasi-isometry classes of horospheres with explicit volume growth calculations.
Estimates growth of reciprocal classes in Hecke groups.
problem Estimating the growth of reciprocal conjugacy classes in Hecke groups.
method Using free product structure and word lengths of reciprocal elements, with tools from basic probability theory.
result Estimates the asymptotic growth of reciprocal conjugacy classes in Hecke groups.
New results on homology torsion growth for various groups.
problem Understanding the growth of higher torsion homologies for arithmetic lattices and other groups.
method Quantitative homotopical method called effective rebuilding, constructing small classifying spaces of finite index subgroups.
result Strong asymptotic bounds for the torsion growth in principal congruence subgroups.
Study shows torsion homology growth vanishes for certain free-by-cyclic groups.
problem Understanding torsion homology growth in free-by-cyclic groups.
method Analyzing polynomially growing monodromy and showing vanishing homology torsion.
result Integral torsion equals ℓ2-torsion for these groups, verifying a conjecture. We provide a lower bound for the uniform exponential growth rate of closed nonflat nonpositively curved 3-manifold groups. A detailed study of the uniform exponential growth rate of closed 3-manifold groups is also presented.
Einstein metrics are blocked by manifold features and group growth.
problem Existence of Einstein metrics on specific 4-manifolds.
method Analysis of collapsing and group growth effects.
result Several 4-manifolds cannot support Einstein metrics due to specific features.
Let G be a virtually special group. Then the residual finiteness growth of G is at most linear. This result cannot be found by embedding G into a special linear group. Indeed, the special linear group SLk(Z), for k>2, has residual finiteness growth nk−1.
Study on RCD(0,N) spaces with small linear diameter growth.
problem Understanding structure properties of RCD(0,N) spaces.
method Analyzing the (revised) fundamental group of RCD(0,N) spaces.
result Proved that the revised fundamental group is finitely generated for RCD(0,N) spaces with small linear diameter growth.
Introduced by Gromov in the nineties, the systolic growth of a Lie group gives the smallest possible covolume of a lattice with a given systole. In a simply connected nilpotent Lie group, this function has polynomial growth, but can grow faster than the volume growth. We express this systolic growth function in terms o…
We prove that the semistability growth of hyperbolic groups is linear, which implies that hyperbolic groups which are sci (simply connected at infinity) have linear sci growth. Based on the linearity of the end-depth of finitely presented groups we show that the linear sci is preserved under amalgamated products over f…
The paper defines higher invariants for groups of polynomial growth and proves their convergence.
problem Defining and proving convergence of higher invariants for groups of polynomial growth.
method Using delocalized cyclic cocycles and a determinant map construction.
result A well-defined pairing between delocalized cyclic cocyles and K-theory classes of C*-algebraic secondary higher invariants.
Strict concavity proven for growth indicator function of certain groups.
problem Proving strict concavity of growth indicator function for specific groups.
method Smoothness of Manhattan hypersurface and critical-exponent map.
result Strict concavity of growth indicator function for relatively Anosov groups.
Fundamental groups of certain Kähler orbifolds have polynomial growth.
problem Understanding the fundamental groups of specific types of orbifolds.
method Analyzing the orbifold fundamental group with respect to the nef anticanonical bundle.
result The orbifold fundamental group has polynomial growth.
Growth rates of geodesics on modular orbifolds are studied.
problem Understanding growth rates of geodesics on modular orbifolds.
method Exhaustion of modular orbifold by compact subsurfaces, analysis of low lying geodesics and reciprocal geodesics.
result Growth rates of low lying geodesics and reciprocal geodesics converge to the full set's growth rate.
Study growth rates of automorphisms of special groups.
problem Understanding the growth rates of automorphisms of special groups.
method Analyzing outer automorphisms of virtually special groups, showing polynomial or exponential growth, and constructing Nielsen-Thurston decompositions.
result Outer automorphism groups of virtually special groups are boundary amenable, have finite virtual cohomological dimension, and satisfy the Tits alternative.
Open manifolds with nonnegative Ricci curvature and Euclidean volume growth have finitely generated fundamental groups.
problem Understanding fundamental groups of open manifolds with specific curvature properties.
method Analyzing universal covers with Euclidean volume growth and applying topological group theory.
result Fundamental groups are finitely generated and virtually abelian under given conditions.
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 G4 and G5 in H4 and H5 have the smallest growth rate. 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 Γn. In this paper, we consider the formal power series whose n-th coefficient is the number of copies of a given finite graph in the ball of radius n centred at the identity element in the Cayley graph of a finitely generated group and call it the growth function. Epstein, Iano-Fletcher and Uri Zwick proved that the growth…
Geometric group theory explores groups through their geometric properties.
problem Understanding groups via geometric properties.
method Cayley and Schreier graphs, ping-pong lemma, quasi-isometries, growth of groups, hyperbolicity.
result Gromov's theorem on groups of polynomial growth and amenability.
We show that every non-decreasing function f:N→N bounded from above by an for some a≥1 can be realized (up to a natural equivalence) as the conjugacy growth function of a finitely generated group. We also construct a finitely generated group G and a subgroup H≤G of index 2 such…
We show that the mapping class group of an orientable finite type surface has uniformly exponential growth, as well as various closely related groups. This provides further evidence that mapping class groups may be linear.
The study connects Kleinian group divergence to random walk recurrence.
problem Understanding the recurrence of random walks on Schreier graphs of Kleinian groups.
method Connecting growth rates of orbits, volume, and Schreier graphs.
result Constructing Kleinian groups of divergence type.
We prove super-quadratic lower bounds for the growth of the filling area function of a certain class of Carnot groups. This class contains groups for which it is known that their Dehn function grows no faster than n2logn. We therefore obtain the existence of (finitely generated) nilpotent groups whose Dehn functio…
Let Mod(S) denote the mapping class group of a compact, orientable surface S. We prove that finitely generated subgroups of Mod(S) which are not virtually abelian have uniform exponential growth with minimal growth rate bounded below by a constant depending only, and necessarily, on S. For the proof, we find in any suc…
We establish new strong lower bounds on the (subnormal) subgroup growth of a large class of groups. This includes the fundamental groups of all finite-volume hyperbolic 3-manifolds and all (free non-abelian)-by-cyclic groups. The lower bound is nearly exponential, which should be compared with the fastest possible subg…
The study of double coset growth in specific groups confirms a conjecture about generic 3-manifolds.
problem Understanding the growth of double cosets in specific groups.
method Generalizing previous work on hyperbolic groups, the study proves the growth of double cosets for certain subgroups.
result The double coset growth of certain subgroups is comparable to the orbital growth function.
Formula for subgroup growth in mapping class groups.
problem Growth of subgroups in mapping class groups.
method Asymptotic growth formula with Teichmüller metric.
result Asymptotic growth formula for subgroups.
Polynomial growth elements found in all subgroups of Out(F_n).
problem Understanding polynomial growth in subgroups of Out(F_n).
method Analyzing conjugacy classes and elements of Out(F_n).
result Polynomial growth elements exist in all subgroups of Out(F_n).
The paper shows how contracting elements in groups lead to large quotients with specific growth rates.
problem Understanding the growth rates of group actions with contracting elements.
method Using extension lemma, rotating families theory, and quasi-tree construction.
result There exist sequences of quotient groups with growth rates approaching the original group's growth rate.
Growth functions of Coxeter groups and the Poincare series of Kleinian and Fuchsian singularities are 2-tangle q-fractions.
Let G be a finitely generated group with a finite generating set S. For g∈G, let lS(g) be the length of the shortest word over S representing g. The growth series of G with respect to S is the series A(t)=∑n=0∞antn, where an is the number of elements of G with lS(g)=n. If…
Study homology growth in nonpositive curvature spaces, finding examples of torsion.
problem Understanding homology growth in nonpositive curvature spaces.
method Computing mod p homology growth of right-angled Artin groups and closed locally CAT(0) manifolds.
result Homology torsion grows exponentially in the index of subgroups, contradicting rational homology growth.
Exponential growth of stable subgroups in Morse geodesics.
problem Growth rates of stable subgroups in complex groups.
method Theory of automatic structures on Morse geodesics.
result Exponential growth of stable subgroups is faster than their infinite index stable subgroups.