We consider analogues of Grigorchuk-Gupta-Sidki (GGS-)groups acting on trees of growing degree; the so-called growing GGS-groups. These groups are not just infinite and do not possess the congruence subgroup property, but many of them are branch and have the p-congruence subgroup property, for a prime p. Among them, we find groups with maximal subgroups only of finite index, and with infinitely many such maximal subgroups. These give the first examples of finitely generated branch groups with infinitely many finite-index maximal subgroups. Additionally, we prove that congruence quotients of growing GGS-groups associated to a defining vector of zero sum give rise to Beauville groups.

We explicitly construct new subgroups of the mapping class groups of an uncountable collection of infinite-type surfaces, including, but not limited to, free groups, Baumslag-Solitar groups, mapping class groups of other surfaces, and a large collection of wreath products. For each such subgroup H and surface S, we show that there are countably many non-conjugate embeddings of H into Map(S); in certain cases, there are uncountably many such embeddings. The images of each of these embeddings cannot lie in the isometry group of S for any hyperbolic metric and are not contained in the closure of the compactly supported subgroup of Map(S). In this sense, our construction is new and does not rely on previously known techniques for constructing subgroups of mapping class groups. Notably, our embeddings of Map(S ') into Map(S) are not induced by embeddings of S ' into S. Our main tool for all of these constructions is the utilization of special homeomorphisms of S called shift maps, and more generally, multipush maps.

We prove a variety of results about subgroups of Thompson's group V. First we prove that every action graph of a finitely generated subgroup of V acting on an orbit in Cantor space is quasi-isometric to a tree. Then we prove that for a broad class of groups of homeomorphisms of the real line, for example Thompson's group F, any action on the Cantor space via an embedding into Thompson's group V must be semiconjugate to the standard action on the line. Finally, we use this to establish that many such groups cannot embed into V; in particular the Stein group F_2,3 cannot embed in V, answering a question of the third author.

Every finite simple group can be generated by two elements and, in fact, every nontrivial element is contained in a generating pair. Groups with this property are said to be 32-generated, and the finite 32-generated groups were recently classified. Turning to infinite groups, in this paper, we prove that the finitely presented simple group T of Thompson is 32-generated. Moreover, we exhibit an element zeta is an element of T such that for any nontrivial alpha is an element of T, there exists gamma is an element of T such that = T.

Every finite simple group can be generated by two elements and, in fact, every nontrivial element is contained in a generating pair. Groups with this property are said to be $${3 \over 2}$$ 3 2 -generated, and the finite $${3 \over 2}$$ 3 2 -generated groups were recently classified. Turning to infinite groups, in this paper, we prove that the finitely presented simple group T of Thompson is $${3 \over 2}$$ 3 2 -generated. Moreover, we exhibit an element ζ ∈ T such that for any nontrivial α ∈ T , there exists γ ∈ T such that 〈 α , ζ γ 〉 = T .

We introduce the concept of a type system P \mathcal {P} , that is, a partition on the set of finite words over the alphabet { 0 , 1 } \{0,1\} compatible with the partial action of Thompson’s group V V , and associate a subgroup Stab V ⁡ ( P ) \operatorname {Stab}_{V}(\mathcal {P}) of V V . We classify the finite simple type systems and show that the stabilizers of various simple type systems, including all finite simple type systems, are maximal subgroups of V V . We also find an uncountable family of pairwise nonisomorphic maximal subgroups of V V . These maximal subgroups occur as stabilizers of infinite simple type systems and have not been described in previous literature: specifically, they do not arise as stabilizers in V V of finite sets of points in Cantor space. Finally, we show that two natural conditions on subgroups of V V (both related to primitivity) are each satisfied only by V V itself, giving new ways to recognise when a subgroup of V V is not actually proper.

Every finite simple group can be generated by two elements and, in fact, every nontrivial element is contained in a generating pair. Groups with this property are said to be $\frac{3}{2}$-generated, and the finite $\frac{3}{2}$-generated groups were recently classified. Turning to infinite groups, in this paper, we prove that the finitely presented simple group $T$ of Thompson is $\frac{3}{2}$-generated. Moreover, we exhibit an element $\zeta \in T$ such that for any nontrivial $\alpha \in T$, there exists $\gamma \in T$ such that $\langle \alpha, \zeta^\gamma \rangle = T$.

We show that continuous epimorphisms between a class of subgroups of mapping class groups of orientable infinite-genus 2-manifolds with no planar ends are always induced by homeomorphisms. This class of subgroups includes the pure mapping class group, the closure of the compactly supported mapping classes, and the full mapping class group in the case that the underlying manifold has a finite number of ends or is perfectly self-similar. As a corollary, these groups are Hopfian topological groups.

We show that R. Thompson’s group T is a maximal subgroup of the group V. The argument provides examples of foundational calculations which arise when expressing elements of V as products of transpositions of basic clopen sets in the Cantor space ℭ .

We generalize the notion of asymptotic mapping class groups and allow them to surject to the Higman--Thompson groups, answering a question of Aramayona and Vlamis in the case of the Higman--Thompson groups. When the underlying surface is a disk, these new asymptotic mapping class groups can be identified with the ribbon and oriented ribbon Higman--Thompson groups. We use this model to prove that the ribbon Higman--Thompson groups satisfy homological stability, providing the first homological stability result for dense subgroups of big mapping class groups. Our result can also be treated as an extension of Szymik--Wahl's work on homological stability for the Higman--Thompson groups to the surface setting.

We provide a family of generating sets S_α of the Higman–Thompson groups V_n that are parametrized by certain sequences α of elements in V_n. These generating sets consist of 3 involutions σ, τ, and s_α, where the latter involution is inspired by the class of spinal elements in the theory of branch groups. In particular this shows the existence of generating sets of V_n that consist of 3 involutions.

We introduce "braided" versions of self-similar groups and Röver–Nekrashevych groups, and study their finiteness properties. This generalizes work of Aroca and Cumplido, and the first author and Wu, who considered the case when the self-similar groups are what we call "self-identical". In particular we use a braided version of the Grigorchuk group to construct a new group called the braided Röver group, which we prove is of type F_∞. Our techniques involve using so called d-ary cloning systems to construct the groups, and analyzing certain complexes of embedded disks in a surface to understand their finiteness properties.

We study the finiteness properties of the braided Higman–Thompson group bV_d,r(H) with labels in H≤ B_d, and bF_d,r(H) and bT_d,r(H) with labels in H≤ PB_d where B_d is the braid group with d strings and PB_d is its pure braid subgroup. We show that for all d≥ 2 and r≥ 1, the group bV_d,r(H) (resp. bT_d,r(H) or bF_d,r(H)) is of type F_n if and only if H is. Our result in particular confirms a recent conjecture of Aroca and Cumplido.

A Cantor surface 𝒞_d is a non-compact surface obtained by gluing copies of a fixed compact surface Y^d (a block), with d+1 boundary components, in a tree-like fashion. For a fixed subgroup H. Moreover, there is a natural surjection onto the family symmetric Thompson groups of Farley–Hughes ; in particular, they provide a positive answer to . We prove that, when the block is a (holed) sphere or a (holed) torus, 𝔅_d(H) is of type F_n if and only if H is of type F_n. As a consequence, for every n, Map(C_d) has a subgroup of type F_n but not F_n+1 which contains the mapping class group of every compact subsurface of 𝒞_d.

We prove that the group of almost-automorphisms of the infinite rooted regular [Formula: see text]-ary tree [Formula: see text] arises naturally as the Thompson-like group of a so-called [Formula: see text]-ary cloning system. A similar phenomenon occurs for any Röver–Nekrashevych group [Formula: see text], for [Formula: see text] a self-similar group. We use this framework to expand on work of Belk and Matucci, who proved that the Röver group, using the Grigorchuk group for [Formula: see text], is of type [Formula: see text]. Namely, we find some natural conditions on subgroups of [Formula: see text] to ensure that [Formula: see text] is of type [Formula: see text] and, in particular, we prove this for all [Formula: see text] in the infinite family of Šunić groups. We also prove that if [Formula: see text] is itself of type [Formula: see text], then so is [Formula: see text], and that every finitely generated virtually free group is self-similar, so in particular every finitely generated virtually free group [Formula: see text] yields a type [Formula: see text] Röver–Nekrashevych group [Formula: see text].

We study the finiteness properties of the braided Higman-Thompson group $bV_{d,r}(H)$ with labels in $H\leq B_d$, and $bF_{d,r}(H)$ and $bT_{d,r}(H)$ with labels in $H\leq PB_d$ where $B_d$ is the braid group with $d$ strings and $PB_d$ is its pure braid subgroup. We show that for all $d\geq 2$ and $r\geq 1$, the group $bV_{d,r}(H)$ (resp. $bT_{d,r}(H)$ or $bF_{d,r}(H)$) is of type $F_n$ if and only if $H$ is. Our result in particular confirms a recent conjecture of Aroca and Cumplido. We then generalize the notion of asymptotic mapping class groups and allow them to surject to the Higman-Thompson groups, answering a question of Aramayona and Vlamis in the case of the Higman-Thompson groups. When the underlying surface is a disk, these new asymptotic mapping class groups can be identified with the ribbon Higman-Thompson groups. We use this model to prove that the ribbon Higman-Thompson groups satisfy homological stability, providing the first homological stability result for dense subgroups of big mapping class groups.

We study the Cantor–Bendixson rank of the space of subgroups for members of a general class of finitely generated self-replicating branch groups. In particular, we show for G either the Grigorchuk group or the Gupta–Sidki 3 group, the Cantor–Bendixson rank of Sub(G) is ω. For each natural number n, we additionally characterize the subgroups of rank n and give a description of subgroups in the perfect kernel.

We construct a family of groups which generalize the Hanoi towers group and study the congruence subgroup problem for the groups in this family. We show that unlike the Hanoi towers group, the groups in this generalization are just infinite and have a trivial rigid kernel. We also put strict bounds on the branch kernel. Additionally, we show that these groups have subgroups of finite index with non-trivial rigid kernel. The only previously known group where this kernel is non-trivial is the Hanoi towers group and so this adds infinitely many new examples. Finally, we show that the topological closures of these groups have Hausdorff dimension arbitrarily close to 1.

We realize lamplighter groups A\wr \mathbb Z , with A a finite abelian group, as automaton groups via affine transformations of power series rings with coefficients in a finite commutative ring. Our methods can realize A\wr \mathbb Z as a bireversible automaton group if and only if the 2-Sylow subgroup of A has no multiplicity one summands in its expression as a direct sum of cyclic groups of order a power of 2.

We show that for every positive integer n there exists a simple group that is of type F_n-1 but not of type F_n . For n≥ 3 these groups are the first known examples of this kind. They also provide infinitely many quasi-isometry classes of finitely presented simple groups. The only previously known infinite family of such classes, due to Caprace–Rémy, consists of non-affine Kac–Moody groups over finite fields. Our examples arise from Röver–Nekrashevych groups, and contain free abelian groups of infinite rank.