Subgroup Of Infinite Index Research Articles

Contracting boundary of a cusped space

Let [Formula: see text] be a finitely generated group. Cashen and Mackay proved that if the contracting boundary of [Formula: see text] with the topology of fellow traveling quasi-geodesics is compact then [Formula: see text] is a hyperbolic group. Let [Formula: see text] be a finite collection of finitely generated infinite index subgroups of [Formula: see text]. Let [Formula: see text] be the cusped space obtained by attaching combinatorial horoballs to each left coset of elements of [Formula: see text]. In this article, we prove that if the combinatorial horoballs are contracting and [Formula: see text] has compact contracting boundary then [Formula: see text] is hyperbolic relative to [Formula: see text].

On maximal subgroups of Thompson’s group $F$

We study subgroups of Thompson’s group F by means of an automaton associated with them. We prove that every maximal subgroup of F of infinite index is closed , that is, it coincides with the subgroup of F accepted by the automaton associated with it. It follows that every finitely generated maximal subgroup of F is undistorted in F . We also prove that every finitely generated subgroup of F is contained in a finitely generated maximal subgroup of F and construct an infinite family of non-isomorphic maximal subgroups of infinite index in F .

Maximal subgroups of a family of iterated monodromy groups

Abstract The Basilica group is a well-known 2-generated weakly branch, but not branch, group acting on the binary rooted tree. Recently, a more general form of the Basilica group has been investigated by Petschick and Rajeev, which is an $s$ -generated weakly branch, but not branch, group that acts on the $m$ -adic tree, for $s,m\ge 2$ . A larger family of groups, which contains these generalised Basilica groups, is the family of iterated monodromy groups. With the new developments by Francoeur, the study of the existence of maximal subgroups of infinite index has been extended from branch groups to weakly branch groups. Here we show that a subfamily of iterated monodromy groups, which more closely resemble the generalised Basilica groups, have maximal subgroups only of finite index.

The Generation Problem in Thompson Group 𝐹

We show that the generation problem in Thompson’s group F F is decidable, i.e., there is an algorithm which decides if a finite set of elements of F F generates the whole F F . The algorithm makes use of the Stallings 2 2 -core of subgroups of F F , which can be defined in an analogous way to the Stallings core of subgroups of a finitely generated free group. Further study of the Stallings 2 2 -core of subgroups of F F provides a solution to another algorithmic problem in F F . Namely, given a finitely generated subgroup H H of F F , it is decidable if H H acts transitively on the set of finite dyadic fractions D \mathcal D . Other applications of the study include the construction of new maximal subgroups of F F of infinite index, among which, a maximal subgroup of infinite index which acts transitively on the set D \mathcal D and the construction of an elementary amenable subgroup of F F which is maximal in a normal subgroup of F F .

Infinite volume and infinite injectivity radius

We prove the following conjecture of Margulis. Let $G$ be a higher rank simple Lie group, and let $\Lambda\le G$ be a discrete subgroup of infinite covolume. Then, the locally symmetric space $\Lambda\backslash G/K$ admits injected balls of any radius. This can be considered as a geometric interpretation of the celebrated Margulis normal subgroup theorem. However, it applies to general discrete subgroups not necessarily associated to lattices. Yet, the result is new even for subgroups of infinite index of lattices. We establish similar results for higher rank semisimple groups with Kazhdan's property (T). We prove a stiffness result for discrete stationary random subgroups in higher rank semisimple groups and a stationary variant of the Stuck--Zimmer theorem for higher rank semisimple groups with property (T). We also show that a stationary limit of a measure supported on discrete subgroups is almost surely discrete.

P-Basilica Groups

We consider a generalisation of the Basilica group to all odd primes: the p-Basilica groups acting on the p-adic tree. We show that the p-Basilica groups have the p-congruence subgroup property but not the congruence subgroup property nor the weak congruence subgroup property. This provides the first examples of weakly branch groups with such properties. In addition, the p-Basilica groups give the first examples of weakly branch, but not branch, groups which are super strongly fractal. We compute the orders of the congruence quotients of these groups, which enable us to determine the Hausdorff dimensions of the p-Basilica groups. Lastly, we show that the p-Basilica groups do not possess maximal subgroups of infinite index and that they have infinitely many non-normal maximal subgroups.

Maximal subgroups of nontorsion Grigorchuk–Gupta–Sidki groups

AbstractA Grigorchuk–Gupta–Sidki (GGS)-group is a subgroup of the automorphism group of the p-regular rooted tree for an odd prime p, generated by one rooted automorphism and one directed automorphism. Pervova proved that all torsion GGS-groups do not have maximal subgroups of infinite index. Here, we extend the result to nontorsion GGS-groups, which include the weakly regular branch, but not branch, GGS-group.

On the oriented Thompson subgroup F→3 and its relatives in higher Brown–Thompson groups

A few years ago the so-called oriented subgroup [Formula: see text] of the Thompson group [Formula: see text] was introduced by V. Jones while investigating the connections between subfactors and conformal field theories. In the coding of links and knots by elements of [Formula: see text] it corresponds exactly to the oriented ones. Thanks to the work of Golan and Sapir, [Formula: see text] provided the first example of a maximal subgroup of infinite index in [Formula: see text] different from the parabolic subgroups that fix a point in [Formula: see text]. In this paper we investigate possible analogues of [Formula: see text] in higher Thompson groups [Formula: see text], with [Formula: see text], introduced by Brown. Most notably, we study algebraic properties of the oriented subgroup [Formula: see text] of [Formula: see text], as described recently by Jones, and prove in particular that it gives rise to a non-parabolic maximal subgroup of infinite index in [Formula: see text] and that the corresponding quasi-regular representation is irreducible.

Irreducibility of a free group endomorphism is a mapping torus invariant

We prove that the property of a free group endomorphism being irreducible is a group invariant of the ascending HNN extension it defines. This answers a question posed by Dowdall–Kapovich–Leininger. We further prove that being irreducible and atoroidal is a commensurability invariant. The invariance follows from an algebraic characterization of ascending HNN extensions that determines exactly when their defining endomorphisms are irreducible and atoroidal; specifically, we show that the endomorphism is irreducible and atoroidal if and only if the ascending HNN extension has no infinite index subgroups that are ascending HNN extensions.

Subgroups of pro-p $$PD ^3$$-groups

We study 3-dimensional Poincaré duality pro-p groups in the spirit of the work by Robert Bieri and Jonathan Hillmann, and show that if such a pro-p group G has a nontrivial finitely presented subnormal subgroup of infinite index, then either the subgroup is cyclic and normal or the subgroup is cyclic and the group is polycyclic or the subgroup is Demushkin and normal in an open subgroup of G. Also, we describe the centralizers of finitely generated subgroups of 3-dimensional Poincaré duality pro-p groups.

On maximal subgroups of infinite index in branch and weakly branch groups

We generalise a technical tool, originally developed by Pervova for the study of maximal subgroups in Grigorchuk and GGS groups, to all weakly branch groups satisfying a natural condition, and in particular to all branch groups. We then use this tool to prove that every maximal subgroup of infinite index of a branch group is also a branch group. As a further application of this result, we show that every maximal subgroup of the Basilica group is of finite index.

Limits of conjugacy classes under iterates of hyperbolic elements of Out($\mathbb F$)

For a free group \mathbb F of finite rank such that rank (\mathbb F) \geq 3 , we prove that the set of weak limits of a conjugacy class in \mathbb F under iterates of some hyperbolic \phi \in \mathrm {Out} (\mathbb F) is equal to the collection of generic leaves and lines with endpoints in attracting fixed points of \phi . As an application we describe the ending lamination set for a hyperbolic extension of \mathbb F by a hyperbolic element of Out( \mathbb F ) in a new way and use it to prove results about Cannon–Thurston maps for such extensions. We also use it to derive conditions for quasiconvexity of finitely generated, infinite index subgroups of \mathbb F in the extension group. These results generalize similar results obtained in [23] and [19] and use different techniques.

On the coarse geometry of certain right-angled Coxeter groups

Let $\Gamma$ be a connected, triangle-free, planar graph with at least five vertices that has no separating vertices or edges. If the graph $\Gamma$ is $\mathcal{CFS}$, we prove that the right-angled Coxeter group $G_\Gamma$ is virtually a Seifert manifold group or virtually a graph manifold group and we give a complete quasi-isometry classification of these such groups. Otherwise, we prove that $G_\Gamma$ is hyperbolic relative to a collection of $\mathcal{CFS}$ right-angled Coxeter subgroups of $G_\Gamma$. Consequently, the divergence of $G_\Gamma$ is linear, or quadratic, or exponential. We also generalize right-angled Coxeter groups which are virtually graph manifold groups to certain high dimensional right-angled Coxeter groups (our families exist in every dimension) and study the coarse geometry of this collection. We prove that strongly quasiconvex torsion free infinite index subgroups in certain graph of groups are free and we apply this result to our right-angled Coxeter groups.

Hybrid lattices and thin subgroups of Picard modular groups

We consider a certain hybridization construction which produces a subgroup of PU(n,1) from a pair of lattices in PU(n−1,1). Among the Picard modular groups PU(2,1,Od), we show that the hybrid of pairs of Fuchsian subgroups PU(1,1,Od) is a lattice when d=1 and d=7, and a geometrically infinite thin subgroup when d=3, that is an infinite-index subgroup with the same Zariski-closure as the full lattice.

Equivariant dissipation in non-archimedean groups

We prove that, if a topological group G has an open subgroup of infinite index, then every net of tight Borel probability measures on G UEB-converging to invariance dissipates in G in the sense of Gromov. In particular, this solves a 2006 problem by Pestov: for every left-invariant (or right-invariant) metric d on the infinite symmetric group Sym(ℕ), compatible with the topology of pointwise convergence, the sequence of the finite symmetric groups (Sym(n), d ↾Sym(n), μSym(n))n∈ℕ equipped with the restricted metrics and their normalized counting measures dissipates, thus fails to admit a subsequence being Cauchy with respect to Gromov’s observable distance.

Stable subgroups and Morse subgroups in mapping class groups

For a finitely generated group, there are two recent generalizations of the notion of a quasiconvex subgroup of a word-hyperbolic group, namely a stable subgroup and a Morse or strongly quasiconvex subgroup. Durham and Taylor [M. Durham and S. Taylor, Convex cocompactness and stability in mapping class groups, Algebr. Geom. Topol. 15(5) (2015) 2839–2859] defined stability and proved stability is equivalent to convex cocompactness in mapping class groups. Another natural generalization of quasiconvexity is given by the notion of a Morse or strongly quasiconvex subgroup of a finitely generated group, studied recently by Tran [H. Tran, On strongly quasiconvex subgroups, To Appear in Geom. Topol., preprint (2017), arXiv:1707.05581 ] and Genevois [A. Genevois, Hyperbolicities in CAT (0) cube complexes, preprint (2017), arXiv:1709.08843 ]. In general, a subgroup is stable if and only if the subgroup is Morse and hyperbolic. In this paper, we prove that two properties of being Morse and stable coincide for a subgroup of infinite index in the mapping class group of an oriented, connected, finite type surface with negative Euler characteristic.

Non-cocompact Group Actions and -Semistability at Infinity

AbstractA finitely presented 1-ended group $G$ has semistable fundamental group at infinity if $G$ acts geometrically on a simply connected and locally compact ANR $Y$ having the property that any two proper rays in $Y$ are properly homotopic. This property of $Y$ captures a notion of connectivity at infinity stronger than “1-ended”, and is in fact a feature of $G$, being independent of choices. It is a fundamental property in the homotopical study of finitely presented groups. While many important classes of groups have been shown to have semistable fundamental group at infinity, the question of whether every $G$ has this property has been a recognized open question for nearly forty years. In this paper we attack the problem by considering a proper but non-cocompact action of a group $J$ on such an $Y$. This $J$ would typically be a subgroup of infinite index in the geometrically acting over-group $G$; for example $J$ might be infinite cyclic or some other subgroup whose semistability properties are known. We divide the semistability property of $G$ into a $J$-part and a “perpendicular to $J$” part, and we analyze how these two parts fit together. Among other things, this analysis leads to a proof (in a companion paper) that a class of groups previously considered to be likely counter examples do in fact have the semistability property.

On strongly quasiconvex subgroups

We develop a theory of \emph{strongly quasiconvex subgroups} of an arbitrary finitely generated group. Strong quasiconvexity generalizes quasiconvexity in hyperbolic groups and is preserved under quasi-isometry. We show that strongly quasiconvex subgroups are also more reflexive of the ambient groups geometry than the stable subgroups defined by Durham-Taylor, while still having many analogous properties to those of quasiconvex subgroups of hyperbolic groups. We characterize strongly quasiconvex subgroups in terms of the lower relative divergence of ambient groups with respect to them. We also study strong quasiconvexity and stability in relatively hyperbolic groups, right-angled Coxeter groups, and right-angled Artin groups. We give complete descriptions of strong quasiconvexity and stability in relatively hyperbolic groups and we characterize strongly quasiconvex special subgroups and stable special subgroups of two dimensional right-angled Coxeter groups. In the case of right-angled Artin groups, we prove that two notions of strong quasiconvexity and stability are equivalent when the right-angled Artin group is one-ended and the subgroups have infinite index. We also characterize non-trivial strongly quasiconvex subgroups of infinite index (i.e. non-trivial stable subgroups) in right-angled Artin groups by quadratic lower relative divergence, expanding the work of Koberda-Mangahas-Taylor on characterizing purely loxodromic subgroups of right-angled Artin groups.

Quotients of the mapping class group by power subgroups

We study the quotient of the mapping class group $\operatorname{Mod}_g^n$ of a surface of genus $g$ with $n$ punctures, by the subgroup $\operatorname{Mod}_g^n[p]$ generated by the $p$-th powers of Dehn twists. Our first main result is that $\operatorname{Mod}_g^1 /\operatorname{Mod}_g^1[p]$ contains an infinite normal subgroup of infinite index, and in particular is not commensurable to a higher-rank lattice, for all but finitely many explicit values of $p$. Next, we prove that $\operatorname{Mod}_g^0/ \operatorname{Mod}_g^0[p]$ contains a K\"ahler subgroup of finite index, for every $p\ge 2$ coprime with six. Finally, we observe that the existence of finite-index subgroups of $\operatorname{Mod}_g^0$ with infinite abelianization is equivalent to the analogous problem for $\operatorname{Mod}_g^0/ \operatorname{Mod}_g^0[p]$.

Maximal subgroups of groups of intermediate growth

Finding the number of maximal subgroups of infinite index of a finitely generated group is a natural problem that has been solved for several classes of “geometric” groups (linear groups, hyperbolic groups, mapping class groups, etc). Here we provide a solution for a family of groups with a different geometric origin: groups of intermediate growth that act on rooted binary trees. In particular, we show that the non-torsion iterated monodromy groups of the tent map (a special case of some groups first introduced by Šunić in [32] as “siblings of the Grigorchuk group”) have exactly countably many maximal subgroups of infinite index, and describe them up to conjugacy. This is in contrast to the torsion case (e.g. Grigorchuk group) where there are no maximal subgroups of infinite index. It is also in contrast to the above-mentioned geometric groups, where there are either none or uncountably many such subgroups.Along the way we show that all the groups defined by Šunić have the congruence subgroup property and are just infinite.