We define a family of discontinuous maps on the circle, called Bowen–Series-like maps, for geometric presentations of surface groups. The family has $2N$ parameters, where $2N$ is the number of generators of the presentation. We prove that all maps in the family have the same topological entropy, which coincides with the volume entropy of the group presentation. This approach allows a simple algorithmic computation of the volume entropy from the presentation only, using the Milnor–Thurston theory for one-dimensional maps.

We give a new criterion which guarantees that a free group admits a bi-ordering that is invariant under a given automorphism. As an application, we show that the fundamental group of the “magic manifold” is bi-orderable, answering a question of Kin and Rolfsen.

Let M be a closed oriented 3-manifold equipped with an Euler structure e and an acyclic representation of its fundamental group. We define a twisted self-linking homology class of the diagonal of the two-point configuration space of M with respect to e. This twisted self-linking homology class appears as an obstruction in the Chern–Simons perturbation theory. When the representation is the maximal free abelian representation $\rho_0$, we prove that our self-linking class is a properly defined “logarithmic derivative” of the Reidemeister–Turaev torsion of $(M,\rho_0,e)$ equipped with the given Euler structure.

We consider a family of cyclic presentations and show that, subject to certain conditions on the defining parameters, they are spines of closed 3-manifolds. These are new examples where the reduced Whitehead graphs are of the same type as those of the Fractional Fibonacci presentations; here the corresponding manifolds are often (but not always) hyperbolic. We also express a lens space construction in terms of a class of positive cyclic presentations that are spines of closed 3-manifolds. These presentations then furnish examples where the Whitehead graphs are of the same type as those of the positive cyclic presentations of type $\mathfrak {Z}$, as considered by McDermott.

We determine the adjoint Reidemeister torsion of a $3$-manifold obtained by some Dehn surgery along K, where K is either the figure-eight knot or the $5_2$-knot. As in a vanishing conjecture (Benini et al. (2020, Journal of High Energy Physics 2020, 57), Gang et al. (2020, Journal of High Energy Physics 2020, 164), and Gang et al. (2021, Advances in Theoretical and Mathematical Physics 25, 1819–1845)), we consider a similar conjecture and show that the conjecture holds for the 3-manifold.

Residual torsion-free nilpotence has proved to be an important property for knot groups with applications to bi-orderability and ribbon concordance. Mayland proposed a strategy to show that a two-bridge knot group has a commutator subgroup which is a union of an ascending chain of para-free groups. This paper proves Mayland’s assertion and expands the result to the subgroups of two-bridge link groups that correspond to the kernels of maps to $\mathbb{Z}$. We call these kernels the Alexander subgroups of the links. As a result, we show the bi-orderability of a large family of two-bridge link groups. This proof makes use of a modified version of a graph-theoretic construction of Hirasawa and Murasugi in order to understand the structure of the Alexander subgroup for a two-bridge link group.

A handlebody link is a union of handlebodies of positive genus embedded in 3-space, which generalises the notion of links in classical knot theory. In this paper, we consider handlebody links with a genus two handlebody and $n-1$ solid tori, $n>1$. Our main result is the classification of such handlebody links with six crossings or less, up to ambient isotopy.

Let G be a group, and let g be a nontrivial element in G. If some nonempty finite product of conjugates of g equals the identity, then g is called a generalized torsion element. We say that a knot K has generalized torsion if $G(K) = \pi _1(S^3 - K)$ admits such an element. For a $(2, 2q+1)$-torus knot K, we demonstrate that there are infinitely many unknots $c_n$ in $S^3$ such that p-twisting K about $c_n$ yields a twist family $\{ K_{q, n, p}\}_{p \in \mathbb {Z}}$ in which $K_{q, n, p}$ is a hyperbolic knot with generalized torsion whenever $|p|> 3$. This gives a new infinite class of hyperbolic knots having generalized torsion. In particular, each class contains knots with arbitrarily high genus. We also show that some twisted torus knots, including the $(-2, 3, 7)$-pretzel knot, have generalized torsion. Because generalized torsion is an obstruction for having bi-order, these knots have non-bi-orderable knot groups.

We generalize results of Thomas, Allcock, Thom–Petersen, and Kar–Niblo to the first $\ell ^{2}$-Betti number of quotients of certain groups acting on trees by subgroups with free actions on the edge sets of the graphs.

Infinite product operations are at the forefront of the study of homotopy groups of Peano continua and other locally path-connected spaces. In this paper, we define what it means for a space X to have infinitely commutative $\pi _1$-operations at a point $x\in X$. Using a characterization in terms of the Specker group, we identify several natural situations in which this property arises. Maintaining a topological viewpoint, we define the transfinite abelianization of a fundamental group at any set of points $A\subseteq X$ in a way that refines and extends previous work on the subject.

We show that every coarse moduli space, parametrizing complex special linear rank-2 local systems with fixed boundary traces on a surface with nonempty boundary, is log Calabi–Yau in that it has a normal projective compactification with trivial log canonical divisor. We connect this to a novel symmetry of generating series for counts of essential multicurves on the surface.

In 1969, Baumslag introduced a family of parafree groups Gi,j which share many properties with the free group of rank 2. The isomorphism problem for the family Gi,j is known to be difficult; a few small partial results have been found so far. In this paper, we compute the twisted Alexander ideals of the groups Gi,j associated with non-abelian representations into $SL(2,{\mathbb Z}_2)$. Using the twisted Alexander ideals, we prove that several pairs of groups among Gi,j are not isomorphic. As a consequence, we solve the isomorphism problem for sub-families containing infinitely many groups Gi,j.

Let $\mathcal{M}$ denote the mapping class group of Σ, a compact connected oriented surface with one boundary component. The action of $\mathcal{M}$ on the nilpotent quotients of π1(Σ) allows to define the so-called Johnson filtration and the Johnson homomorphisms. J. Levine introduced a new filtration of $\mathcal{M}$, called the Lagrangian filtration. He also introduced a version of the Johnson homomorphisms for this new filtration. The first term of the Lagrangian filtration is the Lagrangian mapping class group, whose definition involves a handlebody bounded by Σ, and which contains the Torelli group. These constructions extend in a natural way to the monoid of homology cobordisms. Besides, D. Cheptea, K. Habiro and G. Massuyeau constructed a functorial extension of the LMO invariant, called the LMO functor, which takes values in a category of diagrams. In this paper we give a topological interpretation of the upper part of the tree reduction of the LMO functor in terms of the homomorphisms defined by J. Levine for the Lagrangian mapping class group. We also compare the Johnson filtration with the filtration introduced by J. Levine.

We present a new test for studying asphericity and diagrammatic reducibility of group presentations. Our test can be applied to prove diagrammatic reducibility in cases where the classical weight test fails. We use this criterion to generalize results of J. Howie and S.M. Gersten on asphericity of LOTs and of Adian presentations, and derive new results on solvability of equations over groups. We also use our methods to investigate a conjecture of S.V. Ivanov related to Kaplansky's problem on zero divisors: we strengthen Ivanov's result for locally indicable groups and prove a weak version of the conjecture.

It is known that a bi-orderable group has no generalized torsion element, but the converse does not hold in general. We conjecture that the converse holds for the fundamental groups of 3-manifolds and verify the conjecture for non-hyperbolic, geometric 3-manifolds. We also confirm the conjecture for some infinite families of closed hyperbolic 3-manifolds. In the course of the proof, we prove that each standard generator of the Fibonacci group $F(2,m)\,(m>2)$ is a generalized torsion element.

Excluding four exceptional cases, we determine the asphericity of the relative presentation for m ⩾ 2. If H = ⟨g, h⟩ ⩽ G, then the exceptional cases occur when H is isomorphic to C 5 or C 6.

In this paper, we extend the definition of the $SL\left( 2,\,\mathbb{C} \right)$ Casson invariant to arbitrary knots $K$ in integral homology 3-spheres and relate it to the $m$ -degree of the $\widehat{A}$ -polynomial of $K$ . We prove a product formula for the $\widehat{A}$ -polynomial of the connected sum ${{K}_{1}}\#{{K}_{2}}$ of two knots in ${{S}^{3}}$ and deduce additivity of the $SL\left( 2,\,\mathbb{C} \right)$ Casson knot invariant under connected sums for a large class of knots in ${{S}^{3}}$ . We also present an example of a nontrivial knot $K$ in ${{S}^{3}}$ with trivial $\widehat{A}$ -polynomial and trivial $SL\left( 2,\,\mathbb{C} \right)$ Casson knot invariant, showing that neither of these invariants detect the unknot.

We exhibit infinitely many hyperbolic 3-manifold groups that are not right-orderable.

We prove that certain tree products of finitely generated Abelian groups have Property E. Using this fact, we show that the outer automorphism groups of those tree products of Abelian groups and Brauner’s groups are residually finite.