We construct the first examples of finitely presented groups whose conjugator length function is exponential; these are central extensions of groups of the form $F_m\rtimes F_2$. Further, we use a fiber product construction to exhibit a family of finitely presented groups $\Gamma _k$, where for each k, the conjugator length function of $\Gamma _k$ grows like functions lying in the k-th level of the Grzegorczyk hierarchy of primitive recursive functions.

We completely characterize insoluble groups whose subgroups are pronormal or pronormalize every subgroup. We also completely describe $p$-groups with this property and prove many structural results in the soluble case (for example, such groups are always of length at most $3$ and have Sylow towers).

We consider a random right-angled Coxeter group (RACG), $W_{\Gamma}$, whose presentation graph $\Gamma$ is taken to be an Erdős–Rényi random graph, i.e. $\Gamma\sim\mathcal{G}_{n,p}$. We use techniques from probabilistic combinatorics to establish several new results about the geometry of this random group model. Resolving a conjecture of Susse, we determine the connectivity threshold for square percolation on the random graph $\Gamma\sim\mathcal{G}_{n,p}$. We use this result to determine a large range of p for which the random RACG $W_{\Gamma}$ has a unique cubical coarse median structure. Until Fioravanti et al. (2026), there were no non-hyperbolic examples of groups with cubical coarse rigidity. Perhaps surprisingly, we show that the property of coarse cubical rigidity is in fact typically satisfied by a random RACG for a wide range of the parameter p, including $p=\frac12$, for which the random RACG is also typically non-hyperbolic.

A subgroup X of a group G is called normal-by-finite if it is finite over its core $X_G$. We investigate the behaviour of uncountable groups G of regular cardinality $\aleph $ in which all subgroups of cardinality $\aleph $ are normal-by-finite. Moreover, generalised soluble groups whose subgroups of infinite rank are either normal-by-finite or contranormal are characterised, and the analogous problem is considered for uncountable groups whose subgroups of large cardinality satisfy the same condition.

Given a class $\mathcal{P}$ of groups we say that a group G is fully residually $\mathcal{P}$ if for any finite subset F of G, there exists an epimorphism from G to a group in $\mathcal{P}$ which is injective on F. It is known that any non-trivial knot group is fully residually finite, but not residually free. For hyperbolic knots, its knot group is fully residually closed hyperbolic 3–manifold group, and such a group is fully residually simple. In this paper, we show that every non-trivial knot group is fully residually perfect, closed 3–manifold group.

The landmark quantum complexity result MIP$^*$ $=$ RE has been used to prove the existence of a non-Connes-embeddable tracial von Neumann algebra. Recently, similar ideas have been used to give a negative solution to the Aldous–Lyons conjecture: there is a non-co-sofic invariant random subgroup (IRS) on any non-abelian free group. We define a notion of hyperlinearity for an IRS and show that there is a non-co-hyperlinear IRS on any non-abelian free group. As a corollary, we prove that there is a relation whose von Neumann algebra is not Connes embeddable. We do this by significantly simplifying the reduction of Aldous–Lyons to non-local games, removing the need for subgroup tests entirely.

Given a countable group A with trivial bounded cohomology for all semi-separable coefficients and any group $\Gamma$, we prove that the bounded cohomology of certain restricted verbal wreath products $\Gamma \wr^{_W} A$ vanishes in positive degrees, for all semi-separable coefficients. Examples of such verbal wreath products include the standard restricted wreath product (extending a recent result by Monod for lamplighters groups) as well as verbal wreath products arising from n-solvable, n-nilpotent and k-Burnside ($k = 2,3,4,6$) verbal products. Moreover, we prove that the stable commutator length of any verbal wreath product $\Gamma \wr^{_W} A$ vanishes. As an application, we show that every group of type $F_p$ isometrically embeds into a group of type $F_p$ with vanishing bounded cohomology in degree $n\geq 1$ for all semi-separable coefficients.

In this paper, we construct an implementable algorithm that solves the conjugacy problem in twisted right-angled Artin groups (T-RAAGs). In certain cases, the complexity is known to be linear, by reducing the problem to the twisted conjugacy problem in RAAGs. We also show that T-RAAGs are biautomatic, providing an alternative solution to the conjugacy problem.

In this paper, we introduce the symmetric normaliser graph of a group G. The vertex set of this graph consists of the elements of the group. Vertices x and y are adjacent if x lies in the normaliser of $\langle y \rangle $ and y lies in the normaliser of $\langle x \rangle $. We investigate the position this graph occupies in the hierarchy of graphs defined on groups. We show that the existing hierarchy is further refined by this graph and that the edges of this graph lie between the edges of the commuting graph and the nilpotent graph. For finite groups, we prove a necessary and sufficient condition for the symmetric normaliser graph to be equal to the commuting graph, and similarly, for equality with the nilpotent graph. The edge set of the symmetric normaliser graph is also a subset of the edge set of the Engel graph of a group and has connections to the nongenerating graph of a group.

We give a method to produce faithful representations of the groups $G(n,m)=\langle X, Y \ \vert \ X^m = Y^n \rangle $ in $\mathrm {GL}_2(\mathbb {C}[t^{\pm 1}, q^{\pm 1}])$. These groups are Garside groups and the Garside normal forms of elements of the corresponding monoid can be explicitly recovered from the matrices, in the spirit of Krammer’s proof of the linearity of Artin’s braid groups. We use this method to construct several explicit faithful representations of the above groups, among which a representation which generalizes the reduced Burau representation of $B_3 \cong G(2,3)$ to a large family of groups of the form $G(n,m)$ with n, m coprime (which are torus knot groups). Like the Burau representation, this representation specializes to a representation of a reflection-like quotient that we previously introduced, called 2-toric reflection group. As a byproduct, we get a “Burau representation” for some exceptional complex braid groups, which also shows that the latter embed into their Hecke algebra.

The paper is concerned with maximal subgroups of the ample (better known as topological full) groups of homeomorphisms of totally disconnected compact metrizable topological spaces. We describe all maximal subgroups that are stabilizers of finite sets. Under certain assumptions on the ample group (including minimality), we describe all maximal subgroups that are stabilizers of closed sets or stabilizers of partitions into clopen sets. In particular, our results apply to the ample groups associated with Cantor minimal systems and to some Higman–Thompson groups.

The rational homology of the IA-automorphism group $\operatorname {IA}_n$ of the free group $F_n$ is still mysterious. We study the quotient of the rational homology of $\operatorname {IA}_n$ that is obtained as the image of the map induced by the abelianization map, which we call the Albanese homology of $\operatorname {IA}_n$. We obtain a representation-stable $\operatorname {GL}(n,\mathbb {Q})$-subquotient of the Albanese homology of $\operatorname {IA}_n$, which conjecturally coincides with the entire Albanese homology of $\operatorname {IA}_n$. In particular, we obtain a lower bound of the dimension of the Albanese homology of $\operatorname {IA}_n$ for each homological degree in a stable range. Moreover, we determine the entire third Albanese homology of $\operatorname {IA}_n$ for $n\ge 9$. We also study the Albanese homology of an analogue of $\operatorname {IA}_n$ for the outer automorphism group of $F_n$ and the Albanese homology of the Torelli groups of surfaces. Moreover, we study the relation between the Albanese homology of $\operatorname {IA}_n$ and the cohomology of $\operatorname {Aut}(F_n)$ with twisted coefficients.

Let $F \ast G$ be a free product of a free group F and a LERF group G. We provide sufficient conditions for a subgroup H of $F \ast G$ to be $\mathcal A\cup \mathcal S$-separable, that is, for any finite set $\{\gamma _1, \ldots , \gamma _n\} \subset (F \ast G) \setminus H$, there is a surjection f from $F \ast G$ to an alternating or symmetric group such that $f(\gamma _i) \notin f(H)$ for all i. As a corollary, any finitely generated infinite-index subgroup of a free group is $\mathcal A\cup \mathcal S$-separable in the free product of the free group and an arbitrary LERF group, generalising a result of Wilton [‘Alternating quotients of free groups’, Enseign. Math. 58 (2012), 49–60].

If a complex X is a subcomplex of a diagrammatically reducible (DR) 2-complex Y that has locally indicable fundamental group, then X has locally indicable fundamental group. This is a consequence of the Corson–Trace characterization of diagrammatic reducibility. In this article, we use a Corson–Trace-like characterization of diagrammatic reducibility away from a subcomplex to obtain a considerable stronger result. We apply this to the question of local indicability in the context of Whitehead’s asphericity conjecture. We show that an injective labeled oriented tree that is DR of degree 2, and all its quotients are as well, is locally indicable.

We study the topological dynamics of the action of an acylindrically hyperbolic group on the space of its infinite index convex cocompact subgroups by conjugation. We show that, for any suitable probability measure $\mu $, random walks with respect to $\mu $ will produce elements with strong mixing properties for this action asymptotically almost surely. In particular, when the group has no finite normal subgroups, this implies that the action is highly topologically transitive. Along the way, we prove technical results about convex cocompact subgroups that allow us to extend some results on random walks of Abbott and the first author.

Let M be a pinched negatively curved Riemannian orbifold, whose fundamental group has torsion of order $2$. Generalising results of Sarnak and Erlandsson-Souto for constant curvature oriented surfaces, and with very different techniques, we give an asymptotic counting result on the number of strongly reversible periodic orbits of the geodesic flow in $T^1M$, and prove their equidistribution towards the Bowen-Margulis measure. The result holds in the more general setting with weights coming from thermodynamic formalism, and also in the analogous setting of graphs of groups with $2$-torsion. We give new examples in real hyperbolic Coxeter groups, complex hyperbolic orbifolds and graphs of groups.

We provide a direct connection between the $\mathcal{Z}_{\max}$ (or essential) JSJ decomposition and the Friedl–Tillmann polytope of a hyperbolic two-generator one-relator group with abelianisation of rank 2.

We deduce various structural and algorithmic properties, like the existence of a quadratic-time algorithm computing the $\mathcal{Z}_{\max}$-JSJ decomposition of such groups.

Mapping class groups are known to admit geometric (proper, cobounded) actions on injective spaces. Starting with such an action, and relying only on geometric arguments, we show that all finite generating sets resulting from taking large enough balls in the respective injective space yield word metrics where pseudo-Anosov maps are exponentially generic. We also show that growth tightness holds true for the Cayley graphs corresponding to these finite generating sets, providing a positive answer to a question by Arzhantseva, Cashen and Tao (Growth tight actions, Pacific J. Math. 278 (2015), 1–49).

We describe standard forms for elements of the higher-dimensional Thompson groups nV arising from gridding subdivision processes. These processes lead to standard normal form descriptions for elements in these groups, and sizes of these standard forms estimate the word length with respect to finite generating sets. These gridded forms lead to standard algebraic descriptions as well, with respect to the both infinite and finite generating sets for these groups.

A monoid presentation is called special if the right-hand side of each defining relation is equal to 1. We prove results which relate the two-sided homological finiteness properties of a monoid defined by a special presentation with those of its group of units. Specifically we show that the monoid enjoys the homological finiteness property bi- if its group of units is of type . We also obtain results which relate the Hochschild cohomological dimension of the monoid to the cohomological dimension of its group of units. In particular we show that the Hochschild cohomological dimension of the monoid is bounded above by the maximum of $2$ and the cohomological dimension of its group of units. We apply these results to prove a Lyndon’s Identity type theorem for the two-sided homology of one-relator monoids of the form $\langle A \mid r=1 \rangle $. In particular, we show that all such monoids are of type bi-. Moreover, we show that if r is not a proper power then the one-relator monoid has Hochschild cohomological dimension at most $2$, while if r is a proper power then it has infinite Hochschild cohomological dimension. For any nonspecial one-relator monoid M with defining relation $u=v$ we show that if there is no nonempty word w such that $u,v \in A^*w \cap w A^*$ then M is of type bi-$\mathrm {FP}_\infty $ and has Hochschild cohomological dimension at most $2$.