Let (W, S) be a Coxeter system of rank n, and let $p_{(W, S)}(t)$ be its growth function. It is known that $p_{(W, S)}(q^{-1}) \lt \infty$ holds for all $n \leq q \in \mathbb{N}$. In this paper, we will show that this still holds for $q = n-1$, if (W, S) is 2-spherical. Moreover, we will prove that $p_{(W, S)}(q^{-1}) = \infty$ holds for $q = n-2$, if the Coxeter diagram of (W, S) is the complete graph. These two results provide a complete characterization of the finiteness of the growth function in the case of 2-spherical Coxeter systems with a complete Coxeter diagram.

We study quotients of mapping class groups of punctured spheres by suitable large powers of Dehn twists, showing an analogue of Ivanov’s theorem for the automorphisms of the corresponding quotients of curve graphs. Then we use this result to prove quasi-isometric rigidity of these quotients, answering a question of Behrstock, Hagen, Martin, and Sisto in the case of punctured spheres. Finally, we show that the automorphism groups of our quotients of mapping class groups are “small”, as are their abstract commensurators. This is again an analogue of a theorem of Ivanov about the automorphism group of the mapping class group.

In the process, we develop techniques to extract combinatorial data from a quasi-isometry of a hierarchically hyperbolic space, and use them to give a different proof of a result of Bowditch about quasi-isometric rigidity of pants graphs of punctured spheres.

We show that the group $ \langle a,b,c,t \,:\, a^t=b,b^t=c,c^t=ca^{-1} \rangle$ is profinitely rigid amongst free-by-cyclic groups, providing the first example of a hyperbolic free-by-cyclic group with this property.

Using a recent result of Bowden, Hensel and Webb, we prove the existence of a homeomorphism with positive stable commutator length in the group of homeomorphisms of the Klein bottle which are isotopic to the identity.

We prove several results showing that every locally finite Borel graph whose large-scale geometry is ‘tree-like’ induces a treeable equivalence relation. In particular, our hypotheses hold if each component of the original graph either has bounded tree-width or is quasi-isometric to a tree, answering a question of Tucker-Drob. In the latter case, we moreover show that there exists a Borel quasi-isometry to a Borel forest, under the additional assumption of (componentwise) bounded degree. We also extend these results on quasi-treeings to Borel proper metric spaces. In fact, our most general result shows treeability of countable Borel equivalence relations equipped with an abstract wallspace structure on each class obeying some local finiteness conditions, which we call a proper walling. The proof is based on the Stone duality between proper wallings and median graphs (i.e., CAT(0) cube complexes). Finally, we strengthen the conclusion of treeability in these results to hyperfiniteness in the case where the original graph has one (selected) end per component, generalizing the same result for trees due to Dougherty–Jackson–Kechris.

In a paper from 1980, Shelah constructed an uncountable group all of whose proper subgroups are countable. Assuming the continuum hypothesis, he constructed an uncountable group G that moreover admits an integer n satisfying that for every uncountable $X\subseteq G$, every element of G may be written as a group word of length n in the elements of X. The former is called a Jónsson group, and the latter is called a Shelah group.

In this paper, we construct a Shelah group on the grounds of $\textsf {{ZFC}}$ alone – that is, without assuming the continuum hypothesis. More generally, we identify a combinatorial condition (coming from the theories of negative square-bracket partition relations and strongly unbounded subadditive maps) sufficient for the construction of a Shelah group of size $\kappa $, and we prove that the condition holds true for all successors of regular cardinals (such as $\kappa =\aleph _1,\aleph _2,\aleph _3,\ldots $). This also yields the first consistent example of a Shelah group of size a limit cardinal.

Given a presentation of a monoid $M$, combined work of Pride and of Guba and Sapir provides an exact sequence connecting the relation bimodule of the presentation (in the sense of Ivanov) with the first homology of the Squier complex of the presentation, which is naturally a $\mathbb ZM$-bimodule. This exact sequence was used by Kobayashi and Otto to prove the equivalence of Pride’s finite homological type property with the homological finiteness condition bi-$\mathrm {FP}_3$. Guba and Sapir used this exact sequence to describe the abelianization of a diagram group. We prove here a generalization of this exact sequence of bimodules for presentations of associative algebras. Our proof is more elementary than the original proof for the special case of monoids.

We prove that a homomorphism between free groups of finite rank equipped with the bi-invariant word metrics associated with finite generating sets is a quasi-isometry if and only if it is an isomorphism.

We present a solution to the conjugacy problem in the group of outer automorphisms of $F_3$, a free group of rank 3. We distinguish according to several computable invariants, such as irreducibility, subgroups of polynomial growth and subgroups carrying the attracting lamination. We establish, by considerations on train tracks, that the conjugacy problem is decidable for the outer automorphisms of $F_3$ that preserve a given rank 2 free factor. Then we establish, by consideration on mapping tori, that it is decidable for outer automorphisms of $F_3$ whose maximal polynomial growth subgroups are cyclic. This covers all the cases left by the state of the art.

Stochastic embeddings of finite metric spaces into graph-theoretic trees have proven to be a vital tool for constructing approximation algorithms in theoretical computer science. In the present work, we build out some of the basic theory of stochastic embeddings in the infinite setting with an aim toward applications to Lipschitz free space theory. We prove that proper metric spaces stochastically embedding into $\mathbb {R}$-trees have Lipschitz free spaces isomorphic to $L^1$-spaces. We then undergo a systematic study of stochastic embeddability of Gromov hyperbolic metric spaces into $\mathbb {R}$-trees by way of stochastic embeddability of their boundaries into ultrametric spaces. The following are obtained as our main results: (1) every snowflake of a compact, finite Nagata-dimensional metric space stochastically embeds into an ultrametric space and has Lipschitz free space isomorphic to $\ell ^1$, (2) the Lipschitz free space over hyperbolic n-space is isomorphic to the Lipschitz free space over Euclidean n-space and (3) every infinite, finitely generated hyperbolic group stochastically embeds into an $\mathbb {R}$-tree, has Lipschitz free space isomorphic to $\ell ^1$, and admits a proper, uniformly Lipschitz affine action on $\ell ^1$.

Let Γ be a finite graph and let $A(\Gamma)$ be the corresponding right-angled Artin group. From an arbitrary basis $\mathcal B$ of $H^1(A(\Gamma),\mathbb F)$ over an arbitrary field, we construct a natural graph $\Gamma_{\mathcal B}$ from the cup product, called the cohomology basis graph. We show that $\Gamma_{\mathcal B}$ always contains Γ as a subgraph. This provides an effective way to reconstruct the defining graph Γ from the cohomology of $A(\Gamma)$, to characterize the planarity of the defining graph from the algebra of $A(\Gamma)$ and to recover many other natural graph-theoretic invariants. We also investigate the behaviour of the cohomology basis graph under passage to elementary subminors and show that it is not well-behaved under edge contraction.

Let $(W,S)$ be a Coxeter system, and write $S=\{s_i:i\in I\}$, where I is a finite index set. Fix a nonempty convex subset $\mathscr {L}$ of W. If W is of type A, then $\mathscr {L}$ is the set of linear extensions of a poset, and there are important Bender–Knuth involutions $\mathrm {BK}_i\colon \mathscr {L}\to \mathscr {L}$ indexed by elements of I. For arbitrary W and for each $i\in I$, we introduce an operator $\tau _i\colon W\to W$ (depending on $\mathscr {L}$) that we call a noninvertible Bender–Knuth toggle; this operator restricts to an involution on $\mathscr {L}$ that coincides with $\mathrm {BK}_i$ in type A. Given a Coxeter element $c=s_{i_n}\cdots s_{i_1}$, we consider the operator $\mathrm {Pro}_c=\tau _{i_n}\cdots \tau _{i_1}$. We say W is futuristic if for every nonempty finite convex set $\mathscr {L}$, every Coxeter element c and every $u\in W$, there exists an integer $K\geq 0$ such that $\mathrm {Pro}_c^K(u)\in \mathscr {L}$. We prove that finite Coxeter groups, right-angled Coxeter groups, rank-3 Coxeter groups, affine Coxeter groups of types $\widetilde A$ and $\widetilde C$, and Coxeter groups whose Coxeter graphs are complete are all futuristic. When W is finite, we actually prove that if $s_{i_N}\cdots s_{i_1}$ is a reduced expression for the long element of W, then $\tau _{i_N}\cdots \tau _{i_1}(W)=\mathscr {L}$; this allows us to determine the smallest integer $\mathrm {M}(c)$ such that $\mathrm {Pro}_c^{{\mathrm {M}}(c)}(W)=\mathscr {L}$ for all $\mathscr {L}$. We also exhibit infinitely many non-futuristic Coxeter groups, including all irreducible affine Coxeter groups that are not of type $\widetilde A$, $\widetilde C$, or $\widetilde G_2$.

Motivated by approaches to the word problem for one-relation monoids arising from work of Adian and Oganesian (1987), Guba (1997), and Ivanov, Margolis, and Meakin (2001), we study the submonoid and rational subset membership problems in one-relation monoids and in positive one-relator groups. We give the first known examples of positive one-relator groups with undecidable submonoid membership problem, and we apply this to give the first known examples of one-relation monoids with undecidable submonoid membership problem. We construct several infinite families of one-relation monoids with undecidable submonoid membership problem, including examples that are defined by relations of the form $w=1$ but which are not groups, and examples defined by relations of the form $u=v$ where both of u and v are nonempty. As a consequence, we obtain a classification of the right-angled Artin groups that can arise as subgroups of one-relation monoids. We also give examples of monoids with a single defining relation of the form $aUb = a$ and examples of the form $aUb=aVa$, with undecidable rational subset membership problem. We give a one-relator group defined by a freely reduced word of the form $uv^{-1}$ with $u, v$ positive words, in which the prefix membership problem is undecidable. Finally, we prove the existence of a special two-relator inverse monoid with undecidable word problem, and in which both the relators are positive words. As a corollary, we also find a positive two-relator group with undecidable prefix membership problem. In proving these results, we introduce new methods for proving undecidability of the rational subset membership problem in monoids and groups, including by finding suitable embeddings of certain trace monoids.

The fine curve graph of a surface was introduced by Bowden, Hensel, and Webb as a graph consisting of essential simple closed curves in the surface. Long, Margalit, Pham, Verberne, and Yao proved that the automorphism group of the fine curve graph of a closed orientable surface is isomorphic to the homeomorphism group of the surface. In this paper, based on their argument, we prove that the automorphism group of the fine curve graph of a closed nonorientable surface $N$ of genus $g \geq 4$ is isomorphic to the homeomorphism group of $N$.

We explore the interplay between $\omega $-categoricity and pseudofiniteness for groups, and we conjecture that $\omega $-categorical pseudofinite groups are finite-by-abelian-by-finite. We show that the conjecture reduces to nilpotent p-groups of class 2, and give a proof that several of the known examples of $\omega $-categorical p-groups satisfy the conjecture. In particular, we show by a direct counting argument that for any odd prime p the ($\omega $-categorical) model companion of the theory of nilpotent class 2 exponent p groups, constructed by Saracino and Wood, is not pseudofinite, and that an $\omega $-categorical group constructed by Baudisch with supersimple rank 1 theory is not pseudofinite. We also survey some scattered literature on $\omega $-categorical groups over 50 years.

We prove that virtually free groups are precisely the hyperbolic groups admitting a language of geodesic words containing a unique representative for each group element with bounded triangles. Equivalently, these are exactly the hyperbolic groups for which the model for the Gromov boundary defined by Silva is well defined.

A nontrivial element of a group is a generalized torsion element if some products of its conjugates is the identity. The minimum number of such conjugates is called a generalized torsion order. We provide several restrictions for generalized torsion orders by using the Alexander polynomial.

Let $\Gamma $ be a finitely generated group of matrices over $\mathbb {C}$. We construct an isometric action of $\Gamma $ on a complete $\mathrm {CAT}(0)$ space such that the restriction of this action to any subgroup of $\Gamma $ containing no nontrivial unipotent elements is well behaved. As an application, we show that if M is a graph manifold that does not admit a nonpositively curved Riemannian metric, then any finite-dimensional $\mathbb {C}$-linear representation of $\pi _1(M)$ maps a nontrivial element of $\pi _1(M)$ to a unipotent matrix. In particular, the fundamental groups of such 3-manifolds do not admit any faithful finite-dimensional unitary representations.

Let $\mathbf {G}$ be a connected reductive algebraic group over an algebraically closed field $\Bbbk $ and ${\mathbf B}$ be a Borel subgroup of ${\mathbf G}$. In this paper, we completely determine the composition factors of the permutation module $\mathbb {F}[{\mathbf G}/{\mathbf B}]$ for any field $\mathbb {F}$.

The group of order-preserving automorphisms of a finitely generated Archimedean ordered group of rank $2$ is either infinite cyclic or trivial according as the ratio in $\mathbb {R}$ of the generators of the subgroup is or is not quadratic over $\mathbb {Q}.$ In the case of an Archimedean ordered group of rank $2$ that is not finitely generated, the group of order-preserving automorphisms is free abelian. Criteria determining the rank of this free abelian group are established.