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.
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.
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.
Anosov subgroups' deformations affect limit cones and growth indicators continuously.
problem Understanding continuous changes in Anosov subgroups' effects on limit cones and growth indicators.
method Continuous variation of limit cones and growth indicators under deformations of Anosov subgroups, with convexity assumptions.
result Limit cones and growth indicators vary continuously under deformations of Anosov 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).
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.
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.
Research examines coamenable subgroups in higher rank groups.
problem Investigates coamenable normal subgroups in higher rank groups.
method Analyzes three complementary phenomena in higher rank groups.
result Growth indicators of coamenable subgroups are not preserved but the Riemannian critical exponent remains rigid.
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…
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.
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…
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 paper studies proper discontinuity of actions on Weyl chamber flow spaces.
problem Properly discontinuous actions on Weyl chamber flow spaces for transverse subgroups.
method Analyzes limit sets and quotient spaces, introduces growth indicators and conformal measures.
result Establishes ergodic dichotomy for Weyl chamber flow and introduces new measures.
GLMM trees identify subgroups with different growth patterns in longitudinal data.
problem Identifying subgroups with distinct growth trajectories in longitudinal studies.
method Extended GLMM trees for longitudinal data.
result Extended GLMM trees outperform other methods in accuracy and speed.
New subgroup found in Lie groups with unusual properties.
problem Finding discrete subgroups with specific properties in Lie groups.
method Constructing a specific subgroup of a higher rank Lie group.
result Found a new subgroup that is dense, discrete, non-lattice, and non-tempered.
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.
We determine the factorial growth rate of the number of finite index subgroups of right-angled Artin groups as a function of the index. This turns out to depend solely on the independence number of the defining graph. We also make a conjecture for right-angled Coxeter groups and prove that it holds in a limited setting…
We prove that non-elementary hyperbolic groups grow exponentially more quickly than their infinite index quasiconvex subgroups. The proof uses the classical tools of automatic structures and Perron-Frobenius theory. We also extend the main result to relatively hyperbolic groups and cubulated groups. These extensions us…
The paper bounds growth indicator functions for discrete subgroups in algebraic groups.
problem Bounding growth indicator functions for discrete subgroups in algebraic groups.
method Pointwise bound and equality conditions for growth indicator functions.
result Strict inequalities and equality conditions for growth indicator functions.
We investigate the homology of finite index subgroups G_i of a given finitely presented group G. Specifically, we examine d_p(G_i), which is the dimension of the first homology of G_i, with mod p coefficients. We say that a collection of finite index subgroups {G_i} has linear growth of mod p homology if the infimum of…
This is the second in a series of four papers (with research announcement posted on this arXiv) that together develop a decomposition theory for subgroups of Out(F_n). In this paper we relativize the "Kolchin-type theorem" from the work of Bestvina, Feighn, and Handel on the Tits alternative, which describes a decompos…
Non-normal subgroups of certain groups grow homologically exponentially.
problem Homological torsion growth in non-normal subgroups of specific groups.
method Proving exponential growth of homological torsion in a sequence of non-normal subgroups.
result Exponential homological torsion growth in a sequence of non-normal subgroups.
New space of currents defined for nonabelian free groups and malnormal subgroups.
problem Understanding growth under iteration of outer automorphisms of nonabelian free groups.
method Introducing a new topological space of currents relative to a malnormal subgroup system.
result Currents associated with elements not in conjugates of A are dense in the space of currents relative to A. The study shows subgroup separability conditions for specific groups.
problem Conditions for subgroup separability in free-by-cyclic and deficiency 1 groups.
method Analyzes polynomially growing monodromy and asymptotic probability of random groups.
result Random deficiency 1 groups are not subgroup separable with positive probability.
We investigate the rank gradient and growth of torsion in homology in residually finite groups. As a tool, we introduce a new complexity notion for generating sets, using measured groupoids and combinatorial cost. As an application we prove the vanishing of the above invariants for Farber sequences of subgroups of righ…
Analytic torsion and Reidemeister torsion help show exponential growth of torsion in cohomology.
problem Exponential growth of torsion in the cohomology of arithmetic groups.
method Analytic torsion and Reidemeister torsion, applied to fibered cusp ends of manifolds.
result Exponential growth of torsion in the cohomology of arithmetic groups.
New CAT(0) groups show superexponential subgroup Dehn functions.
problem Understanding subgroups of CAT(0) groups with superexponential Dehn functions.
method Construction of specific 4- and 6-dimensional CAT(0) groups.
result Found subgroups with Dehn functions exp(n)(xm) and exp(n)(xα). The study shows exponential distortion in virtually special groups containing free subgroups.
problem Understanding distortion in virtually special groups containing free subgroups.
method Constructing examples of virtually special groups with finite rank free subgroups.
result Distortion functions grow like exp^k(x^m) and can be superexponential.
Let Γ be a (non-elementary) convex co-compact group of isometries of a pinched Hadamard manifold X. We show that a normal subgroup Γ0 has critical exponent equal to the critical exponent of Γ if and only if Γ/Γ0 is amenable. We prove a similar result for the exponential growth rate of closed geodesics on $…
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.
We give a new proof of Gromov's theorem that any finitely generated group of polynomial growth has a finite index nilpotent subgroup. Unlike the original proof, it does not rely on the Montgomery-Zippin-Yamabe structure theory of locally compact groups.
We classify isometries of compact Lorentz manifolds.
problem Understanding the isometry group of compact Lorentz manifolds.
method Proving structure theorems and applying the Tits alternative.
result Classification of lattices acting on compact Lorentz manifolds.
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 prove exponential growth rate of contractible closed geodesics for an arbitrary bumpy metric on manifolds of the form X#Y, where the fundamental group of X has a subgroup of finite index at least 3 and Y is simply connected and not a homotopy sphere.
In this paper we prove that for suitable sequences of congruence subgroups of Bianchi groups, including the standard exhaustive sequences of a congruence subgroup, and even symmetric powers of the standard representation of Sl_2(C) the size of the torsion part in the first homology grows exponentially. This extends res…
Study on hyperbolic groups, focusing on separability and splittings.
problem Coarse separability and splittings in hyperbolic groups.
method Quantitative analysis of volume growth and cut-sets, focusing on thickened spheres.
result One-ended hyperbolic groups that are not virtually surface groups are coarsely separable by a subset of subexponential growth if and only if they split over a virtually cyclic subgroup.
We study the growth of the order of torsion subgroups of the homology in a tower of finite abelian coverings. In particular, we prove that it is exponential for when the tower converges to the maximal free abelian cover of a link complement when the first nonzero Alexander polynomial has positive logarithmic Mahler mea…
Affirmative answer to splitting question for open manifolds with nonnegative Ricci curvature and linear volume growth.
problem Understanding the structure of universal covers of open manifolds with nonnegative Ricci curvature and linear volume growth.
method Proving the universal cover splits off an isometric R-factor. result If an open manifold with nonnegative Ricci curvature has linear volume growth, its universal cover is isometric to a metric product RkimesN. 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.
Study on homology growth and ℓ2-Betti numbers of Out(W_n).
problem Growth of homology groups and ℓ2-Betti numbers in Out(W_n). method Farber sequences, Lück's approximation theorem, Gaboriau and Noûs argument, recent method on complex of partial bases.
result Sublinear growth of homology and ℓ2-Betti numbers up to a certain degree, vanishing in some cases. Let l be a link of d components. For every finite-index lattice in Z^d there is an associated finite abelian cover of S^3 branched over l. We show that the order of the torsion subgroup of the first homology of these covers has exponential growth rate equal to the logarithmic Mahler measure of the Alexander polynomial …
Study 2-complexes' homology properties and torsion growth.
problem Quantitative connections between 1-cycle filling inequalities and homology complexities.
method Geometric lower bounds on first homology of finite covers.
result Geometric lower bound on first homology size of finite covers.
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 give a new, effective proof of the separability of cubically convex-cocompact subgroups of special groups. As a consequence, we show that if G is a virtually compact special hyperbolic group, and Q≤G is a K-quasiconvex subgroup, then any g∈G−Q of word-length at most n is separated from Q by a subg…
Let p and l be two distinct prime numbers and let G be a group. We study the asymptotic behaviour of the mod-l Betti numbers in p-adic analytic towers of finite index subgroups. If X is a finite l-group of automorphisms of G, our main theorem allows to lift lower bounds for the mod-l cohomology growth in the fixed poin…
The study examines growth of quadratic forms under Anosov subgroups.
problem Growth of quadratic forms under Anosov subgroups.
method Analyzes exponential bounds and asymptotic counting functions for distances between geodesic copies of symmetric spaces.
result Shows asymptotic behavior of counting functions for certain choices of quadratic forms.
Study on geodesics and dihedral groups in lattices.
problem Growth and distribution of conjugacy classes of dihedral subgroups.
method Generalizing earlier work on reciprocal geodesics, proving equidistribution.
result Reciprocal geodesics are equidistributed in the unit tangent bundle.