Subgroup Of Infinite Index Research Articles

The Surface Group Conjecture: Cyclically Pinched and Conjugacy Pinched One-Relator Groups

The general surface group conjecture asks whether a one-relator group where every subgroup of finite index is again one-relator and every subgroup of infinite index is free (property IF) is a surface group. We resolve several related conjectures given in Fine et al. (Sci Math A 1:1–15, 2008). First we obtain the Surface Group Conjecture B for cyclically pinched and conjugacy pinched one-relator groups. That is: if G is a cyclically pinched one-relator group or conjugacy pinched one-relator group satisfying property IF then G is free, a surface group or a solvable Baumslag–Solitar Group. Further combining results in Fine et al. (Sci Math A 1:1–15, 2008) on Property IF with a theorem of Wilton (Geom Topol, 2012) and results of Stallings (Ann Math 2(88):312–334, 1968) and Kharlampovich and Myasnikov (Trans Am Math Soc 350(2):571–613, 1998) we show that Surface Group Conjecture C proposed in Fine et al. (Sci Math A 1:1–15, 2008) is true, namely: If G is a finitely generated nonfree freely indecomposable fully residually free group with property IF, then G is a surface group.

SIMPLY INTERSECTING PAIR MAPS IN THE MAPPING CLASS GROUP

The Torelli group, [Formula: see text], is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface. There are three types of elements that naturally arise in studying [Formula: see text]: bounding pair maps, separating twists, and simply intersecting pair maps (SIP-maps). Historically the first two types of elements have been the focus of the literature on [Formula: see text], while SIP-maps have received relatively little attention until recently, due to an infinite presentation of [Formula: see text] introduced by Putman that uses all three types of elements. We will give a topological characterization of the image of an SIP-map under the Johnson homomorphism and Birman–Craggs–Johnson homomorphism. We will also classify which SIP-maps are in the kernel of these homomorphisms. Then we will look at the subgroup generated by all SIP-maps, SIP (Sg), and show it is an infinite index subgroup of [Formula: see text].

Pattern rigidity and the Hilbert–Smith conjecture

In this paper we initiate a study of the topological group $PPQI(G,H)$ of pattern-preserving quasi-isometries for $G$ a hyperbolic Poincare duality group and $H$ an infinite quasiconvex subgroup of infinite index in $G$. Suppose $\partial G$ admits a visual metric $d$ with $dim_H < dim_t +2$, where $dim_H$ is the Hausdorff dimension and $dim_t$ is the topological dimension of $(\partial G,d)$. a) If $Q_u$ is a group of pattern-preserving uniform quasi-isometries (or more generally any locally compact group of pattern-preserving quasi-isometries) containing $G$, then $G$ is of finite index in $Q_u$. b) If instead, $H$ is a codimension one filling subgroup, and $Q$ is any group of pattern-preserving quasi-isometries containing $G$, then $G$ is of finite index in $Q$. Moreover, (Topological Pattern Rigidity) if $L$ is the limit set of $H$, $\LL$ is the collection of translates of $L$ under $G$, and $Q$ is any pattern-preserving group of {\it homeomorphisms} of $\partial G$ preserving $\LL$ and containing $G$, then the index of $G$ in $Q$ is finite. We find analogous results in the realm of relative hyperbolicity, regarding an equivariant collection of horoballs as a symmetric pattern in a {\it hyperbolic} (not relatively hyperbolic) space. Combining our main result with a theorem of Mosher-Sageev-Whyte, we obtain QI rigidity results. An important ingredient of the proof is a version of the Hilbert-Smith conjecture for certain metric measure spaces, which uses the full strength of Yang's theorem on actions of the p-adic integers on homology manifolds. This might be of independent interest.

Alternating quotients of free groups

We strengthen Marshall Hall's theorem to show that free groups are locally extended residually alternating. Let F be any free group of rank at least two, let H be a finitely generated subgroup of infinite index in F and let \{\gamma_1,\ldots,\gamma_n\}\subseteq F\smallsetminus H be a finite subset. Then there is a surjection f from F to a finite alternating group such that f(\gamma_i)\notin f(H) for any i . The techniques of this paper can also provide symmetric quotients.

One-ended subgroups of graphs of free groups with cyclic edge groups

Consider a one-ended word-hyperbolic group. If it is the fundamental group of a graph of free groups with cyclic edge groups then either it is the fundamental group of a surface or it contains a finitely generated one-ended subgroup of infinite index. As a corollary, the same holds for limit groups. We also obtain a characterisation of surfaces with boundary among free groups equipped with peripheral structures.

Lagrangian mapping class groups from a group homological point of view

We focus on two kinds of infinite index subgroups of the mapping class group of a surface associated with a Lagrangian submodule of the first homology of a surface. These subgroups, called Lagrangian mapping class groups, are known to play important roles in the interaction between the mapping class group and finite-type invariants of 3-manifolds. In this paper, we discuss these groups from a group (co)homological point of view. The results include the determination of their abelianizations, lower bounds of the second homology and remarks on the (co)homology of higher degrees. As a by-product of this investigation, we determine the second homology of the mapping class group of a surface of genus 3.

Recubulating free groups

We show that for each finitely generated infinite index subgroup H of a free group F, there is a proper cocompact action of F on a CAT(0) cube complex with a single orbit of hyperplane stabilized by conjugates of H.

Groups with many abelian subgroups

It is known that a (generalized) soluble group whose proper subgroups are abelian is either abelian or finite, and finite minimal non-abelian groups are classified. Here we describe the structure of groups in which every subgroup of infinite index is abelian.

Strong approximation in the Apollonian group

The Apollonian group is a finitely generated, infinite index subgroup of the orthogonal group O Q ( Z ) fixing the Descartes quadratic form Q. For nonzero v ∈ Z 4 satisfying Q ( v ) = 0 , the orbits P v = A v correspond to Apollonian circle packings in which every circle has integer curvature. In this paper, we specify the reduction of primitive orbits P v mod any integer d > 1 . We show that this reduction has a multiplicative structure, and that mod primes p ⩾ 5 it is the full cone of integer solutions to Q ( v ) ≡ 0 for v ≢ 0 . This analysis is an essential ingredient in applications of the affine linear sieve as developed by Bourgain, Gamburd and Sarnak.

On subfactors arising from asymptotic representations of symmetric groups

We consider the infinite symmetric group and its infinite index subgroup given as the stabilizer subgroup of one element under the natural action on a countable set. This inclusion of discrete groups induces a hyperfinite subfactor for each finite factorial representation of the larger group. We compute subfactor invariants of this construction in terms of the Thoma parameter.

Normal automorphisms of free Burnside groups

We prove that for an arbitrary odd and every automorphism of the free Burnside group that stabilizes every maximal normal subgroup of infinite index is an inner automorphism. For the same values of and , we establish that the subgroup of inner automorphisms of is maximal among the subgroups in which the orders of the elements are bounded by .

Subgroups of profinite surface groups

We study the subgroup structure of the etale fundamental group $\Pi$ of a projective curve over an algebraically closed field of characteristic 0. We obtain an analog of the diamond theorem for $\Pi$. As a consequence we show that most normal subgroups of infinite index are semi-free. In particular every proper open subgroup of a normal subgroup of infinite index is semi-free.

Finite generation of iterated wreath products

Let (G n , X n ) be a sequence of finite transitive permutation groups with uniformly bounded number of generators. We prove that the infinitely iterated permutational wreath product $${\ldots\wr G_2\wr G_1}$$ is topologically finitely generated if and only if the profinite abelian group $${\prod_{n\geq 1} G_n/G'_n}$$ is topologically finitely generated. As a corollary, for a finite transitive group G the minimal number of generators of the wreath power $${G\wr \ldots\wr G\wr G}$$ (n times) is bounded if G is perfect, and grows linearly if G is non-perfect. As a by-product we construct a finitely generated branch group, which has maximal subgroups of infinite index.

ON GROUPS SATISFYING THE MAXIMAL AND THE MINIMAL CONDITIONS FOR SUBNORMAL SUBGROUPS OF INFINITE ORDER OR INDEX

In this article we will prove that a generalized radical group satisfying the maximal condition for subnormal subgroups of infinite order (the minimal condition for subnormal subgroups of infinite index, respectively) is soluble-by-finite. Such result generalizes that obtained by D. H. Paek in [5].

On Popa's Cocycle Superrigidity Theorem

Theorem 1.1 (Popa’s Cocycle Superrigity, Special Case). Let Γ be a discrete group with Kazhdan’s property (T), let Γ0 < Γ be an infinite index subgroup, (X0,μ0) an arbitrary probability space, and let Γ (X,μ) = (X0,μ0)0 be the corresponding generalized Bernoulli action. Then for any discrete countable group Λ and any measurable cocycle α : Γ × X → Λ there exist a homomorphism : Γ → Λ and a measurable map φ : X → Λ so that:

Lower bounds for resonances of infinite-area Riemann surfaces

For infinite area, geometrically finite surfaces X = Γ\H, we prove new lower bounds on the local density of resonances D(z) when z lies in a logarithmic neighborhood of the real axis. These lower bounds involve the dimension δ of the limit set of Γ. The first bound is valid when δ > 1 2 and shows logarithmic growth of the number D(z) of resonances at high energy i.e. when |Re(z)| → +∞. The second bound holds for δ > 3 4 and if Γ is an infinite index subgroup of certain arithmetic groups. In this case we obtain a polynomial lower bound. Both results are in favor of a conjecture of Guillope-Zworski on the existence of a fractal Weyl law for resonances.

Description of periodic p-harmonic functions on Cayley tree

We show that any periodic (with respect to normal subgroups of finite index of the group representation of the Cayley tree) p-harmonic function on a Cayley tree is a constant. For some normal subgroups of infinite index we describe a class of (non-constant) periodic p-harmonic functions. We also prove that linear combinations of the p-harmonic functions described for normal subgroups of infinite index are also p-harmonic.

On pro-p analogues of limit groups via extensions of centralizers

We begin a study of a pro-p analogue of limit groups via extensions of centralizers and call \({\mathcal{L}}\) this new class of pro-p groups. We show that the pro-p groups of \({\mathcal{L}}\) have finite cohomological dimension, type FP∞ and non-positive Euler characteristic. Among the group theoretic properties it is proved that they are free-by-(torsion free nilpotent) and if non-abelian do not have a finitely generated non-trivial normal subgroup of infinite index. Furthermore it is shown that every 2 generated pro-p group in the class \({\mathcal{L}}\) is either free pro-p or abelian.

Schreier rewriting beyond the classical setting

Using actions of free monoids and free associative algebras, we establish some Schreiertype formulas involving ranks of actions and ranks of subactions in free actions or Grassmann-type relations for the ranks of intersections of subactions of free actions. The coset action of the free group is used to establish a generalization of the Schreier formula in the case of subgroups of infinite index. We also study and apply large modules over free associative and free group algebras.

Large groups of deficiency 1

We prove that if a group possesses a deficiency 1 presentation where one of the relators is a commutator, then it is ℤ × ℤ, large or is as far as possible from being residually finite. Then we use this to show that a mapping torus of an endomorphism of a finitely generated free group is large if it contains a ℤ × ℤ subgroup of infinite index, as well as showing that such a group is large if it contains a Baumslag-Solitar group of infinite index and has a finite index subgroup with first Betti number at least 2. We give applications to free by cyclic groups, 1 relator groups and residually finite groups.