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.

We completely determine finite abelian regular branched covers of the 2-sphere $S^{2}$ with the property that each homeomorphism of $S^{2}$ preserving the branching set can be lifted.

Every Anosov flow on a 3-manifold is associated to a bifoliated plane (a plane endowed with two transverse foliations $F^s$ and $F^u$ ) which reflects the normal structure of the flow endowed with the center-stable and center-unstable foliations. A flow is $\mathbb{R}$ -covered if $F^s$ (or equivalently $F^u$ ) is trivial. On the other hand, from any Anosov flow one can build infinitely many others by Dehn–Goodman–Fried surgeries. This paper investigates how these surgeries modify the bifoliated plane. We first observe that surgeries along orbits corresponding to disjoint simple closed geodesics do not affect the bifoliated plane of the geodesic flow of a hyperbolic surface (Theorem 1). Analogously, for any non- $\mathbb{R}$ -covered Anosov flow, surgeries along pivot periodic orbits do not affect the branching structure of its bifoliated plane (Theorem 2). Next, we consider the set $\mathcal{S}urg(A)$ of Anosov flows obtained by Dehn–Goodman–Fried surgeries from the suspension flow $X_A$ of any hyperbolic matrix $A \in SL(2,\mathbb{Z})$ . Fenley proved that performing only positive (or negative) surgeries on $X_A$ leads to $\mathbb{R}$ -covered Anosov flows. We study here Anosov flows obtained by a combination of positive and negative surgeries on $X_A$ . Among other results, we build non- $\mathbb{R}$ -covered Anosov flows on hyperbolic manifolds. Furthermore, we show that given any flow $X\in \mathcal{S}urg(A)$ there exists $\epsilon>0$ such that every flow obtained from $X$ by a non-trivial surgery along any $\epsilon$ -dense periodic orbit $\gamma$ is $\mathbb{R}$ -covered (Theorem 4). Analogously, for any flow $X \in \mathcal{S}urg(A)$ there exist periodic orbits $\gamma_+,\gamma_-$ such that every flow obtained from $X$ by surgeries with distinct signs on $\gamma_+$ and $\gamma_-$ is non- $\mathbb{R}$ -covered (Theorem 5).

In this paper we give a complete description of the Bieri–Neumann–Strebel–Renz invariants of the Lodha–Moore groups. The second author previously computed the first two invariants, and here we show that all the higher invariants coincide with the second one, which finishes the complete computation. As a consequence, we present a complete picture of the finiteness properties of normal subgroups of the first Lodha–Moore group. In particular, we show that every finitely presented normal subgroup of the group is of type $\textrm{F}_\infty$ , answering a question posed in Oberwolfach in 2018. The proof involves applying a variation of Bestvina–Brady discrete Morse theory to the so called cluster complex X introduced by the first author. As an application, we also demonstrate that a certain simple group S previously constructed by the first author is of type $\textrm{F}_\infty$ . This provides the first example of a type $\textrm{F}_\infty$ simple group that acts faithfully on the circle by homeomorphisms, but does not admit any nontrivial action by $C^1$ -diffeomorphisms, nor by piecewise linear homeomorphisms, on any 1-manifold.

Let G be the group $\text {PAff}_{+}(\mathbb R/\mathbb Z)$ of piecewise affine circle homeomorphisms or the group ${\operatorname {\mathrm {Diff}}}^{{\kern1pt}\infty }(\mathbb R/\mathbb Z)$ of smooth circle diffeomorphisms. A constructive proof that all irrational rotations are distorted in G is given.

In this paper we apply Conley index theory in a covering space of an invariant set S, possibly not isolated, in order to describe the dynamics in S. More specifically, we consider the action of the covering translation group in order to define a topological separation of S which distinguishes the connections between the Morse sets within a Morse decomposition of S. The theory developed herein generalizes the classical connection matrix theory, since one obtains enriched information on the connection maps for non-isolated invariant sets, as well as for isolated invariant sets. Moreover, in the case of an infinite cyclic covering induced by a circle-valued Morse function, one proves that the Novikov differential of f is a particular case of the p-connection matrix defined herein.

Recall that two geodesics in a negatively curved surface S are of the same type if their free homotopy classes differ by a homeomorphism of the surface. In this note we study the distribution in the unit tangent bundle of the geodesics of fixed type, proving that they are asymptotically equidistributed with respect to a certain measure ${\mathfrak {m}}^S$ on $T^1S$ . We study a few properties of this measure, showing for example that it distinguishes between hyperbolic surfaces.

Given groups $A$ and $B$, what is the minimal commutator length of the 2020th (for instance) power of an element $g\in A*B$ not conjugate to elements of the free factors? The exhaustive answer to this question is still unknown, but we can give an almost answer: this minimum is one of two numbers (simply depending on $A$ and $B$). Other similar problems are also considered.

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.

Given an integer $g>2$ , we state necessary and sufficient conditions for a finite Abelian group to act as a group of automorphisms of some compact nonorientable Riemann surface of genus g. This result provides a new method to obtain the symmetric cross-cap number of Abelian groups. We also compute the least symmetric cross-cap number of Abelian groups of a given order and solve the maximum order problem for Abelian groups acting on nonorientable Riemann surfaces.

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.

The Bollobás–Riordan (BR) polynomial [(2002), Math. Ann.323 81] is a universal polynomial invariant for ribbon graphs. We find an extension of this polynomial for a particular family of combinatorial objects, called rank 3 weakly coloured stranded graphs. Stranded graphs arise in the study of tensor models for quantum gravity in physics, and generalize graphs and ribbon graphs. We present a seven-variable polynomial invariant of these graphs, which obeys a contraction/deletion recursion relation similar to that of the Tutte and BR polynamials. However, it is defined on a much broader class of objects, and furthermore captures properties that are not encoded by the Tutte or BR polynomials.

Using a result of Vdovina, we may associate to each complete connected bipartite graph $\kappa $ a two-dimensional square complex, which we call a tile complex, whose link at each vertex is $\kappa $ . We regard the tile complex in two different ways, each having a different structure as a $2$ -rank graph. To each $2$ -rank graph is associated a universal $C^{\star }$ -algebra, for which we compute the K-theory, thus providing a new infinite collection of $2$ -rank graph algebras with explicit K-groups. We determine the homology of the tile complexes and give generalisations of the procedures to complexes and systems consisting of polygons with a higher number of sides.

In the early 1980s, Johnson defined a homomorphism $\mathcal {I}_{g}^1\to \bigwedge ^3 H_1\left (S_{g},\mathbb {Z}\right )$, where $\mathcal {I}_{g}^1$ is the Torelli group of a closed, connected, and oriented surface of genus g with a boundary component and $S_g$ is the corresponding surface without a boundary component. This is known as the Johnson homomorphism.

We study the map induced by the Johnson homomorphism on rational homology groups and apply it to abelian cycles determined by disjoint bounding-pair maps, in order to compute a large quotient of $H_n\left (\mathcal {I}_{g}^1,\mathbb {Q}\right )$ in the stable range. This also implies an analogous result for the stable rational homology of the Torelli group $\mathcal {I}_{g,1}$ of a surface with a marked point instead of a boundary component. Further, we investigate how much of the image of this map is generated by images of such cycles and use this to prove that in the pointed case, they generate a proper subrepresentation of $H_n\left (\mathcal {I}_{g,1}\right )$ for $n\ge 2$ and g large enough.

We define a new ribbon group action on ribbon graphs that uses a semidirect product of a permutation group and the original ribbon group of Ellis-Monaghan and Moffatt to take (partial) twists and duals, or twuals, of ribbon graphs. A ribbon graph is a fixed point of this new ribbon group action if and only if it is isomorphic to one of its (partial) twuals. This extends the original ribbon group action, which only used the canonical identification of edges, to the more natural setting of self-twuality up to isomorphism. We then show that every ribbon graph has in its orbit an orientable embedded bouquet and prove that the (partial) twuality properties of these bouquets propagate through their orbits. Thus, we can determine (partial) twualities via these one vertex graphs, for which checking isomorphism reduces simply to checking dihedral group symmetries. Finally, we apply the new ribbon group action to generate all self-trial ribbon graphs on up to seven edges, in contrast with the few, large, very high-genus, self-trial regular maps found by Wilson, and by Jones and Poultin. We also show how the automorphism group of a ribbon graph yields self-dual, -petrial or –trial graphs in its orbit, and produce an infinite family of self-trial graphs that do not arise as covers or parallel connections of regular maps, thus answering a question of Jones and Poulton.

Let $\operatorname {\mathrm {{\rm G}}}(n)$ be equal to either $\operatorname {\mathrm {{\rm PO}}}(n,1),\operatorname {\mathrm {{\rm PU}}}(n,1)$ or $\operatorname {\mathrm {\textrm {PSp}}}(n,1)$ and let $\Gamma \leq \operatorname {\mathrm {{\rm G}}}(n)$ be a uniform lattice. Denote by $\operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}}$ the hyperbolic space associated to $\operatorname {\mathrm {{\rm G}}}(n)$ , where $\operatorname {\mathrm {{\rm K}}}$ is a division algebra over the reals of dimension d. Assume $d(n-1) \geq 2$ .

In this article we generalise natural maps to measurable cocycles. Given a standard Borel probability $\Gamma $ -space $(X,\mu _X)$ , we assume that a measurable cocycle $\sigma :\Gamma \times X \rightarrow \operatorname {\mathrm {{\rm G}}}(m)$ admits an essentially unique boundary map $\phi :\partial _\infty \operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}} \times X \rightarrow \partial _\infty \operatorname {\mathrm {\mathbb {H}^m_{{\rm K}}}}$ whose slices $\phi _x:\operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}} \rightarrow \operatorname {\mathrm {\mathbb {H}^m_{{\rm K}}}}$ are atomless for almost every $x \in X$ . Then there exists a $\sigma $ -equivariant measurable map $F: \operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}} \times X \rightarrow \operatorname {\mathrm {\mathbb {H}^m_{{\rm K}}}}$ whose slices $F_x:\operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}} \rightarrow \operatorname {\mathrm {\mathbb {H}^m_{{\rm K}}}}$ are differentiable for almost every $x \in X$ and such that $\operatorname {\mathrm {\textrm {Jac}}}_a F_x \leq 1$ for every $a \in \operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}}$ and almost every $x \in X$ . This allows us to define the natural volume $\operatorname {\mathrm {\textrm {NV}}}(\sigma )$ of the cocycle $\sigma $ . This number satisfies the inequality $\operatorname {\mathrm {\textrm {NV}}}(\sigma ) \leq \operatorname {\mathrm {\textrm {Vol}}}(\Gamma \backslash \operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}})$ . Additionally, the equality holds if and only if $\sigma $ is cohomologous to the cocycle induced by the standard lattice embedding $i:\Gamma \rightarrow \operatorname {\mathrm {{\rm G}}}(n) \leq \operatorname {\mathrm {{\rm G}}}(m)$ , modulo possibly a compact subgroup of $\operatorname {\mathrm {{\rm G}}}(m)$ when $m>n$ .

Given a continuous map $f:M \rightarrow N$ between compact hyperbolic manifolds, we also obtain an adaptation of the mapping degree theorem to this context.

Let G be a finitely generated group that can be written as an extension

$$ \begin{align*} 1 \longrightarrow K \stackrel{i}{\longrightarrow} G \stackrel{f}{\longrightarrow} \Gamma \longrightarrow 1 \end{align*} $$

$b_1(G)> b_1(\Gamma ) > 0$

$\Bbb {Z}$

$F \hookrightarrow X \rightarrow B$

$a(X) = 2$

$va(X) = 2$

$1$

We construct the first examples of normal subgroups of mapping class groups that are isomorphic to non-free right-angled Artin groups. Our construction also gives normal, non-free right-angled Artin subgroups of other groups, such as braid groups and pure braid groups, as well as many subgroups of the mapping class group, such as the Torelli subgroup. Our work recovers and generalizes the seminal result of Dahmani–Guirardel–Osin, which gives free, purely pseudo-Anosov normal subgroups of mapping class groups. We give two applications of our methods: (1) we produce an explicit proper normal subgroup of the mapping class group that is not contained in any level $m$ congruence subgroup and (2) we produce an explicit example of a pseudo-Anosov mapping class with the property that all of its even powers have free normal closure and its odd powers normally generate the entire mapping class group. The technical theorem at the heart of our work is a new version of the windmill apparatus of Dahmani–Guirardel–Osin, which is tailored to the setting of group actions on the projection complexes of Bestvina–Bromberg–Fujiwara.

In this paper, we study distance one surgeries between lens spaces L(p, 1) with p ≥ 5 prime and lens spaces L(n, 1) for $$n \in \mathbb{Z}$$ and band surgeries from T (2, p) to T (2, n). In particular, we prove that L(n, 1) is obtained by a distance one surgery from L(5, 1) only if n=±1, 4, ±5, 6 or ±9, and L(n, 1) is obtained by a distance one surgery from L(7, 1) if and only if n=±1, 3, 6, 7, 8 or 11.

We address the question of determining which mapping class groups of infinite-type surfaces admit nonelementary continuous actions on hyperbolic spaces.

More precisely, let $\Sigma $ be a connected, orientable surface of infinite type with tame endspace whose mapping class group is generated by a coarsely bounded subset. We prove that ${\mathrm {Map}}(\Sigma )$ admits a continuous nonelementary action on a hyperbolic space if and only if $\Sigma $ contains a finite-type subsurface which intersects all its homeomorphic translates.

When $\Sigma $ contains such a nondisplaceable subsurface K of finite type, the hyperbolic space we build is constructed from the curve graphs of K and its homeomorphic translates via a construction of Bestvina, Bromberg and Fujiwara. Our construction has several applications: first, the second bounded cohomology of ${\mathrm {Map}}(\Sigma )$ contains an embedded $\ell ^1$; second, using work of Dahmani, Guirardel and Osin, we deduce that ${\mathrm {Map}} (\Sigma )$ contains nontrivial normal free subgroups (while it does not if $\Sigma $ has no nondisplaceable subsurface of finite type), has uncountably many quotients and is SQ-universal.