We compare the marked length spectra of some pairs of proper and cocompact cubical actions of a nonvirtually cyclic group on $\mathrm {CAT}(0)$ cube complexes. The cubulations are required to be virtually co-special, have the same sets of convex-cocompact subgroups, and admit a contracting element. There are many groups for which these conditions are always fulfilled for any pair of cubulations, including nonelementary cubulable hyperbolic groups, many cubulable relatively hyperbolic groups, and many right-angled Artin and Coxeter groups.

For these pairs of cubulations, we study the Manhattan curve associated to their combinatorial metrics. We prove that this curve is analytic and convex, and a straight line if and only if the marked length spectra are homothetic. The same result holds if we consider invariant combinatorial metrics in which the lengths of the edges are not necessarily one. In addition, for their standard combinatorial metrics, we prove a large deviations theorem with shrinking intervals for their marked length spectra. We deduce the same result for pairs of word metrics on hyperbolic groups.

The main tool is the construction of a finite-state automaton that simultaneously encodes the marked length spectra of both cubulations in a coherent way, in analogy with results about (bi)combable functions on hyperbolic groups by Calegari-Fujiwara [14]. The existence of this automaton allows us to apply the machinery of thermodynamic formalism for suspension flows over subshifts of finite type, from which we deduce our results.

Let $G = K \rtimes \langle t \rangle $ be a finitely generated group where K is abelian and $\langle t\rangle$ is the infinite cyclic group. Let R be a finite symmetric subset of K such that $S = \{ (r,1),(0,t^{\pm 1}) \mid r \in R \}$ is a generating set of G. We prove that the spherical conjugacy ratio, and hence the conjugacy ratio, of G with respect to S is 0 unless G is virtually abelian, confirming a conjecture of Ciobanu, Cox and Martino in this case. We also show that the Baumslag–Solitar group $\mathrm{BS}(1,k)$, $k\geq 2$, has a one-sided Følner sequence F such that the conjugacy ratio with respect to F is non-zero, even though $\mathrm{BS}(1,k)$ is not virtually abelian. This is in contrast to two-sided Følner sequences, where Tointon showed that the conjugacy ratio with respect to a two-sided Følner sequence is positive if and only if the group is virtually abelian.

We strengthen the maximal ergodic theorem for actions of groups of polynomial growth to a form involving jump quantity, which is the sharpest result among the family of variational or maximal ergodic theorems. As two applications, we first obtain the upcrossing inequalities with exponential decay of ergodic averages and then provide an explicit bound on the convergence rate such that the ergodic averages with strongly continuous regular group actions are metastable (or locally stable) on a large interval. Before exploiting the transference techniques, we actually obtain a stronger result—the jump estimates on a metric space with a measure not necessarily doubling. The ideas or techniques involve martingale theory, non-doubling Calderón–Zygmund theory, almost orthogonality argument, and some delicate geometric argument involving the balls and the cubes on a group equipped with a not necessarily doubling measure.

Let Fn be the free group on $n \geq 2$ generators. We show that for all $1 \leq m \leq 2n-3$ (respectively, for all $1 \leq m \leq 2n-4$), there exists a subgroup of ${\operatorname{Aut}(F_n)}$ (respectively, ${\operatorname{Out}(F_n)}$), which has finiteness of type Fm but not of type $FP_{m+1}(\mathbb{Q})$; hence, it is not m-coherent. In both cases, the new result is the upper bound $m= 2n-3$ (respectively, $m = 2n-4$), as it cannot be obtained by embedding direct products of free noncyclic groups, and certifies higher incoherence up to the virtual cohomological dimension and is therefore sharp. As a tool of the proof, we discuss the existence and nature of multiple inequivalent extensions of a suitable finite-index subgroup K4 of ${\operatorname{Aut}(F_2)}$ (isomorphic to the quotient of the pure braid group on four strands by its centre): the fibre of four of these extensions arise from the strand-forgetting maps on the braid groups, while a fifth is related with the Cardano–Ferrari epimorphism.

Given a symmetric monoidal category ${\mathcal C}$ with product $\sqcup $, where the neutral element for the product is an initial object, we consider the poset of $\sqcup $-complemented subobjects of a given object X. When this poset has finite height, we define decompositions and partial decompositions of X which are coherent with $\sqcup $, and order them by refinement. From these posets, we define complexes of frames and partial bases, augmented Bergman complexes and related ordered versions. We propose a unified approach to the study of their combinatorics and homotopy type, establishing various properties and relations between them. Via explicit homotopy formulas, we will be able to transfer structural properties, such as Cohen-Macaulayness.

In well-studied scenarios, the poset of $\sqcup $-complemented subobjects specializes to the poset of free factors of a free group, the subspace poset of a vector space, the poset of nondegenerate subspaces of a vector space with a nondegenerate form, and the lattice of flats of a matroid. The decomposition and partial decomposition posets, the complex of frames and partial bases together with the ordered versions, either coincide with well-known structures, generalize them, or yield new interesting objects. In these particular cases, we provide new results along with open questions and conjectures.

We prove that Thompson’s group T and, more generally, all the Higman–Thompson groups $T_n$ have quadratic Dehn function.

The Hanna Neumann conjecture (HNC) for a free group G predicts that $\overline{\chi}(U\cap V)\leqslant \overline{\chi} (U)\overline{\chi}(V)$ for all finitely generated subgroups U and V, where $\overline{\chi}(H) = \max\{-\chi(H),0\}$ denotes the reduced Euler characteristic of H. A strengthened version of the HNC was proved independently by Friedman and Mineyev in 2011. Recently, Antolín and Jaikin-Zapirain introduced the $L^2$-Hall property and showed that if G is a hyperbolic limit group that satisfies this property, then G satisfies the HNC. Antolín and Jaikin-Zapirain established the $L^2$-Hall property for free and surface groups, which Brown and Kharlampovich extended to all limit groups. In this paper, we prove the $L^2$-Hall property for graphs of free groups with cyclic edge groups that are hyperbolic relative to virtually abelian subgroups. We also give another proof of the $L^2$-Hall property for limit groups. As a corollary, we show that all these groups satisfy a strengthened version of the HNC.

It is a theorem due to F. Haglund and D. Wise that reflection groups (aka Coxeter groups) virtually embed into right-angled reflection groups (aka right-angled Coxeter groups). In this article, we generalize this observation to rotation groups, which can be thought of as a common generalization of Coxeter groups and graph products of groups. More precisely, we prove that rotation groups (aka periagroups) virtually embed into right-angled rotation groups (aka graph products of groups).

We study a family of Thompson-like groups built as rearrangement groups of fractals introduced by Belk and Forrest in 2019, each acting on a Ważewski dendrite. Each of these is a finitely generated group that is dense in the full group of homeomorphisms of the dendrite (studied by Monod and Duchesne in 2019) and has infinite-index finitely generated simple commutator subgroup, with a single possible exception. More properties are discussed, including finite subgroups, the conjugacy problem, invariable generation and existence of free subgroups. We discuss many possible generalisations, among which we find the Airplane rearrangement group $T_A$. Despite close connections with Thompson’s group F, dendrite rearrangement groups seem to share many features with Thompson’s group V.

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.

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.

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 $N$.

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 $H\le F$ be two finitely generated free groups. Given $g\in F$, we study the ideal $\mathfrak I_g$ of equations for g with coefficients in H, i.e. the elements $w(x)\in H*\langle x\rangle$ such that $w(g)=1$ in F. The ideal $\mathfrak I_g$ is a normal subgroup of $H*\langle x\rangle$, and it’s possible to algorithmically compute a finite normal generating set for $\mathfrak I_g$; we give a description of one such algorithm, based on Stallings folding operations. We provide an algorithm to find an equation in w(x)\in$\mathfrak I_g$ with minimum degree, i.e.  such that its cyclic reduction contains the minimum possible number of occurrences of x and x−1; this answers a question of A. Rosenmann and E. Ventura. More generally, we show how to algorithmically compute the set Dg of all integers d such that $\mathfrak I_g$ contains equations of degree d; we show that Dg coincides, up to a finite set, with either $\mathbb N$ or $2\mathbb N$. Finally, we provide examples to illustrate the techniques introduced in this paper. We discuss the case where ${\text{rank}}(H)=1$. We prove that both kinds of sets Dg can actually occur. We show that the equations of minimum possible degree aren’t in general enough to generate the whole ideal $\mathfrak I_g$ as a normal subgroup.

We show that when a finitely presented Bestvina–Brady group splits as an amalgamated product over a subgroup $H$, its defining graph contains an induced separating subgraph whose associated Bestvina–Brady group is contained in a conjugate of $H$.