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For every natural number $n$, there exist finitely presented groups with residual finiteness depths $\omega\cdot n$ and $\omega\cdot n + 1$. The ordinals that arise as the residual finiteness depth of a finitely generated group (equivalently, a countable group) are $0,\, 1$, the countable limit ordinals, and the successors of these limit ordinals.

A free-by-cyclic group $F_N\rtimes_\phi\mathbb{Z}$ has non-trivial centre if and only if $[\phi]$ has finite order in ${\rm{Out}}(F_N)$. We establish a profinite ridigity result for such groups: if $\Gamma_1$ is a free-by-cyclic group with non-trivial centre and $\Gamma_2$ is a finitely generated free-by-cyclic group with the same finite quotients as $\Gamma_1$, then $\Gamma_2$ is isomorphic to $\Gamma_1$. One-relator groups with centre are similarly rigid. We prove that finitely generated free-by-(finite cyclic) groups are profinitely rigid in the same sense; the proof revolves around a finite poset $\mathbf{fsc}(G)$ that carries information about the centralisers of finite subgroups of $G$ -- it is a complete invariant for these groups. These results provide contrasts with the lack of profinite rigidity among surface-by-cyclic groups and (free abelian)-by-cyclic groups, as well as general virtually-free groups., Comment: 29 pages, 6 figures. Comments welcome!

Every countable group $G$ can be embedded in a finitely generated group $G^*$ that is hopfian and complete, i.e. $G^*$ has trivial centre and every epimorphism $G^*\to G^*$ is an inner automorphism. Every finite subgroup of $G^*$ is conjugate to a finite subgroup of $G$. If $G$ has a finite presentation (respectively, a finite classifying space), then so does $G^*$. Our construction of $G^*$ relies on the existence of closed hyperbolic 3-manifolds that are asymmetric and non-Haken., Comment: 9 pages, 1 figure. Dedicated to Chuck Miller. To appear in the Bulletin of the Australian Mathematical Society

Given an arbitrary, finitely presented, residually finite group $\Gamma$, one can construct a finitely generated, residually finite, free-by-free group $M_\Gamma = F_\infty\rtimes F_4$ and an embedding $M_\Gamma \hookrightarrow (F_4\ast \Gamma)\times F_4$ that induces an isomorphism of profinite completions. In particular, there is a free-by-free group whose profinite completion contains $\widehat{\Gamma}$ as a retract., Comment: 3 page note. Final version. To appear in Quarterly Journal of Mathematics

Let $\Phi$ be a pseudo-Anosov diffeomorphism of a compact (possibly non-orientable) surface $\Sigma$ with one boundary component. We show that if $b \in \pi_1(\Sigma)$ is the boundary word, $\phi \in {\rm{Aut}}(\pi_1(\Sigma))$ is a representative of $\Phi$ fixing $b$, and ${\rm{ad}}_b$ denotes conjugation by $b$, then the orbits of $\langle \phi, {\rm{ad}}_b \rangle\cong\mathbb{Z}^2$ in the graph of free factors of $\pi_1(\Sigma)$ are quasi-isometrically embedded. It follows that for $N \geq 2$ the free factor graph for ${\rm{Aut}}(F_N)$ is not hyperbolic, in contrast to the ${\rm{Out}}(F_N)$ case., Comment: 12 pages, 1 figure. To appear in GGD

We give a relatively self-contained proof that if a group $G$ fibres algebraically and is part of a $\mathrm{PD}^3$-pair, then $G$ is the fundamental group of a fibred compact aspherical 3-manifold. This yields a homological proof of a classical theorem of Stallings: if $G = \pi_1(M^3)$ is the fundamental group of a compact irreducible 3-manifold $M^3$ and $\phi \colon G \to \mathbb{Z}$ is a surjective homomorphism with finitely generated kernel, then $\phi$ is induced by a topological fibration of $M^3$ over the circle., Comment: 7 pages. v2, added further background to the introduction and a clarification in the proof of Theorem 2. To appear in "Topology at Infinity of Discrete Groups", Contemporary Mathematics series

We establish the following non-abelian analogue of the Fundamental Theorem of Projective Geometry: the natural map from ${\rm{Aut}}(F_n)$ to the automorphism group of the free-factor complex $\mathcal{AF}_n$ is an isomorphism. We also prove the corresponding theorem for the action of ${\rm{Out}}(F_n)$ on the complex of conjugacy classes of free factors.

We describe a flexible construction that produces triples of finitely generated, residually finite groups $M\hookrightarrow P \hookrightarrow \Gamma$, where the maps induce isomorphisms of profinite completions $\widehat{M}\cong\widehat{P}\cong\widehat{\Gamma}$, but $M$ and $\Gamma$ have Serre's property FA while $P$ does not. In this construction, $P$ is finitely presented and $\Gamma$ is of type ${\rm{F}}_\infty$. More generally, given any positive integer $d$, one can demand that $M$ and $\Gamma$ have a fixed point whenever they act by semisimple isometries on a complete CAT$(0)$ space of dimension at most $d$, while $P$ acts without a fixed point on a tree., Comment: 10 pages, no figures

The group $\mathfrak{X}(G)$ is obtained from $G\ast G$ by forcing each element $g$ in the first free factor to commute with the copy of $g$ in the second free factor. We make significant additions to the list of properties that the functor $\mathfrak{X}$ is known to preserve. We also investigate the geometry and complexity of the word problem for $\mathfrak{X}(G)$. Subtle features of $\mathfrak{X}$ are encoded in a normal abelian subgroup $W<\mathfrak{X}(G)$ that is a module over $\mathbb{Z} Q$, where $Q= H_1(G,\mathbb{Z})$. We establish a structural result for this module and illustrate its utility by proving that $\mathfrak{X}$ preserves virtual nilpotence, the Engel condition, and growth type -- polynomial, exponential, or intermediate. We also use it to establish isoperimetric inequalities for $\mathfrak{X}(G)$ when $G$ lies in a class that includes Thompson's group $F$ and all non-fibered K\"ahler groups. The word problem is solvable in $\mathfrak{X}(G)$ if and only if it is solvable in $G$. The Dehn function of $\mathfrak{X}(G)$ is bounded below by a cubic polynomial if $G$ maps onto a non-abelian free group., Comment: 25 pages, no figures. Final version. Accepted for publication in Commentarii Mathematici Helvetici

We explore an elementary construction that produces finitely presented groups with diverse homological finiteness properties -- the {\em binary subgroups}, $B(\Sigma,\mu)

Let $\Gamma$ be a non-elementary Kleinian group and $H<\Gamma$ a finitely generated, proper subgroup. We prove that if $\Gamma$ has finite co-volume, then the profinite completions of $H$ and $\Gamma$ are not isomorphic. If $H$ has finite index in $\Gamma$, then there is a finite group onto which $H$ maps but $\Gamma$ does not. These results streamline the existing proofs that there exist full-sized groups that are profinitely rigid in the absolute sense. They build on a circleof ideas that can be used to distinguish among the profinite completions of subgroups of finite index in other contexts, e.g. limit groups. We construct new examples of profinitely rigid groups, including the fundamental group of the hyperbolic $3$-manifold ${\rm{Vol}}(3)$ and of the $4$-fold cyclic branched cover of the figure-eight knot. We also prove that if a lattice in ${\rm{PSL}}(2,\mathbb{C})$ is profinitely rigid, then so is its normalizer in ${\rm{PSL}}(2,\mathbb{C})$., Comment: Final version. To be published in a special issue of the Michigan Math. J. honoring Gopal Prasad

Leighton's Theorem states that if there is a tree $T$ that covers two finite graphs $G_1$ and $G_2$, then there is a finite graph $\hat G$ that is covered by $T$ and covers both $G_1$ and $G_2$. We prove that this result does not extend to regular covers by graphs other than trees. Nor does it extend to non-regular covers by a quasitree, even if the automorphism group of the quasitree contains a uniform lattice. But it does extend to regular coverings by quasitrees., Comment: 29 pages, 10 figures; v2: minor changes made following referee's comments; v3: minor edits to proof of Theorem 1.1 and correction of Example 2.2; to appear in Algebraic & Geometric Topology

We present novel constructions concerning the homology of finitely generated groups. Each construction draws on ideas of Gilbert Baumslag. There is a finitely presented acyclic group $U$ such that $U$ has no proper subgroups of finite index and every finitely presented group can be embedded in $U$. There is no algorithm that can determine whether or not a finitely presentable subgroup of a residually finite, biautomatic group is perfect. For every recursively presented abelian group $A$ there exists a pair of groups $i:P_A\hookrightarrow G_A$ such that $i$ induces an isomorphism of profinite completions, where $G_A$ is a torsion-free biautomatic group that is residually finite and superperfect, while $P_A$ is a finitely generated group with $H_2(P_A,\mathbb{Z})\cong A$., Comment: Final version, to appear in volume in honour of Baumslag: "Elementary Theory of Group Rings, and Related Topics"

There does not exist an algorithm that can determine whether or not a group presented by commutators is a right-angled Artin group., Comment: Final version. To appear in the Glasgow Mathematical Journal

We address the question: for which collections of finite simple groups does there exist an algorithm that determines the images of an arbitrary finitely presented group that lie in the collection? We prove both positive and negative results. For a collection of finite simple groups that contains infinitely many alternating groups, or contains classical groups of unbounded dimensions, we prove that there is no such algorithm. On the other hand, for families of simple groups of Lie type of bounded rank, we obtain positive results. For example, given any fixed untwisted Lie type $X$ there is an algorithm that determines whether or not an arbitrary finitely presented group has infinitely many simple images isomorphic to $X(q)$ for some $q$, and if there are finitely many, the algorithm determines them.

Direct powers of perfect groups admit more concise presentations than one might naively suppose. If $H_1G=H_2G=0$, then $G^n$ has a presentation with $O(\log n)$ generators and $O(\log n)^3$ relators. If, in addition, there is an element $g\in G$ that has infinite order in every non-trivial quotient of $G$, then $G^n$ has a presentation with $d(G) +1$ generators and $O(\log n)$ relators. The bounds that we obtain on the deficiency of $G^n$ are not monotone in $n$; this points to potential counterexamples for the Relation Gap Problem., Comment: Final version. To appear in Proceedings of the American Mathematical Society. (9 pages, no figures)

We consider the group $\mathfrak{X}(G)$ obtained from $G\ast G$ by forcing each element $g$ in the first free factor to commute with the copy of $g$ in the second free factor. Deceptively complicated finitely presented groups arise from this construction: $\mathfrak{X}(G)$ is finitely presented if and only if $G$ is finitely presented, but if $F$ is a non-abelian free group of finite rank then $\mathfrak{X}(F)$ has a subgroup of finite index whose third homology is not finitely generated., Comment: 12 pages, 2 figures. Final version to appear in Bulletin of the London Math Soc

We construct classes of K\"ahler groups that do not have finite classifying spaces and are not commensurable to subdirect products of surface groups. Each of these groups is the fundamental group of the generic fibre of a holomorphic map from a product of Kodaira fibrations onto an elliptic curve., Comment: 21 pages, 1 figure, V2: Minor corrections and improvements to the exposition. Final accepted version, to appear in the Transactions of the American Mathematical Society

If $M$ is a compact 3-manifold whose first betti number is 1, and $N$ is a compact 3-manifold such that $\pi_1N$ and $\pi_1M$ have the same finite quotients, then $M$ fibres over the circle if and only if $N$ does. We prove that groups of the form $F_2\rtimes\mathbb{Z}$ are distinguished from one another by their profinite completions. Thus, regardless of betti number, if $M$ and $N$ are punctured torus bundles over the circle and $M$ is not homeomorphic to $N$, then there is a finite group $G$ such that one of $\pi_1M$ and $\pi_1N$ maps onto $G$ and the other does not., Comment: 17 pages, no figures. v2 minor corrections. This is the final version accepted for publication

We show that for every finitely presented pro-$p$ nilpotent-by-abelian-by-finite group $G$ there is an upper bound on $\dim_{\mathbb{Q}_p} (H_1(M, \mathbb{Z}_p) \otimes_{\mathbb{Z}_p} \mathbb{Q}_p )$, as $M$ runs through all pro-$p$ subgroups of finite index in $G$., Comment: Final accepted version. To appear in the Israel Journal of Mathematics

We establish results concerning the profinite completions of 3-manifold groups. In particular, we prove that the complement of the figure-eight knot $S^3-K$ is distinguished from all other compact 3-manifolds by the set of finite quotients of its fundamental group. In addition, we show that if $M$ is a compact 3-manifold with $b_1(M)=1$, and $\pi_1(M)$ has the same finite quotients as a free-by-cyclic group $F_r\rtimes\mathbb{Z}$, then $M$ has non-empty boundary, fibres over the circle with compact fibre, and $\pi_1(M)\cong F_r\rtimes_\psi\mathbb{Z}$ for some $\psi\in{\rm{Out}}(F_r)$., Comment: 15 pages, no figures

Motivated by problems in topology, we explore the complexity of balanced group presentations. We obtain large lower bounds on the complexity of Andrews-Curtis trivialisations, beginning in rank 4. Our results are based on a new understanding of how Dehn functions of groups behave under certain kinds of push-outs. We consider groups $S$ with presentations of deficiency 1 satisfying certain technical conditions and construct balanced group presentations $\P_w$ indexed by words $w$ in the generators of $S$. If $w=1$ in $S$ then $\P_w$ is Andrews-Curtis trivialisable and the number of Andrews-Curtis moves required to trivialise it can be bounded above and below in terms of how hard it is to prove that $w=1$ in $S$., Comment: 25 pages, 2 figures

We show that if a group G is finitely presented and nilpotent-by-abelian-by-finite, then there is an upper bound on the first betti number of M as M runs through all subgroups of finite index in G., Comment: 11 pages, no figures. To appear in Pacific Journal of Math

We address the question of whether the property of being virtually special (in the sense of Haglund and Wise) is algorithmically decidable for finite, non-positively curved cube complexes. Our main theorem shows that it cannot be decided locally, i.e. by examining one hyperplane at a time. Specifically, we prove that there does not exist an algorithm that, given a compact non-positively squared 2-complex X and a hyperplane H in X can decide whether or not there is a finite-sheeted cover of X in which no lift of H self-osculates., Comment: 9 pages, 2 figures. This is the final version accepted for publication

A permutoid is a set of partial permutations that contains the identity and is such that partial compositions, when defined, have at most one extension in the set. In 2004 Peter Cameron conjectured that there can exist no algorithm that determines whether or not a finite permutoid based on a finite set can be completed to a finite permutation group, and he related this problem to the study of groups that have no non-trivial finite quotients. This note explains how our recent work on the profinite triviality problem for finitely presented groups can be used to prove Cameron's conjecture. We also prove that the existence problem for finite developments of rigid pseudogroups is unsolvable., Comment: 18 pages, including an appendix by Benjamin Steinberg. This is the final version accepted for publication

We construct the first example of a finitely-presented, residually-finite group that contains an infinite sequence of non-isomorphic finitely-presented subgroups such that each of the inclusion maps induces an isomorphism of profinite completions., Comment: 10 pages, no figures. Final version to appear in Journal of the European Math. Soc

Let $\C(\Gamma)$ be the set of isomorphism classes of the finite groups that are homomorphic images of $\Gamma$. We investigate the extent to which $\C(\Gamma)$ determines $\Gamma$ when $\Gamma$ is a group of geometric interest. If $\Gamma_1$ is a lattice in ${\rm{PSL}}(2,\R)$ and $\Gamma_2$ is a lattice in any connected Lie group, then $\C(\Gamma_1) = \C(\Gamma_2)$ implies that $\Gamma_1$ is isomorphic to $\Gamma_2$. If $F$ is a free group and $\Gamma$ is a right-angled Artin group or a residually free group (with one extra condition), then $\C(F)=\C(\Gamma)$ implies that $F\cong\Gamma$. If $\Gamma_1<{\rm{PSL}}(2,\Bbb C)$ and $\Gamma_2< G$ are non-uniform arithmetic lattices, where $G$ is a semi-simple Lie group with trivial centre and no compact factors, then $\C(\Gamma_1)= \C(\Gamma_2)$ implies that $G \cong {\rm{PSL}}(2,\Bbb C)$ and that $\Gamma_2$ belongs to one of finitely many commensurability classes. These results are proved using the theory of profinite groups; we do not exhibit explicit finite quotients that distinguish among the groups in question. But in the special case of two non-isomorphic triangle groups, we give an explicit description of finite quotients that distinguish between them., Comment: Minor edits. Version accepted by Israel J Math

We consider pairs of finitely presented, residually finite groups $u:P\hookrightarrow \Gamma$. We prove that there is no algorithm that, given an arbitrary such pair, can determine whether or not the associated map of profinite completions $\hat{u}: \widehat{P} \to \widehat{\Gamma}$ is an isomorphism. Nor do there exist algorithms that can decide whether $\hat{u}$ is surjective, or whether $\widehat{P}$ is isomorphic to $\widehat{\Gamma}$., Comment: 12 pages

We prove that there is no algorithm that can determine whether or not a finitely presented group has a non-trivial finite quotient; indeed, this remains undecidable among the fundamental groups of compact, non-positively curved square complexes. We deduce that many other properties of groups are undecidable. For hyperbolic groups, there cannot exist algorithms to determine largeness, the existence of a linear representation with infinite image (over any infinite field), or the rank of the profinite completion., Comment: 36 pages, 2 figures. Final version as accepted for publication, incorporating referee's comments. To appear in Inventiones Mathematicae

We study the asymptotic growth of homology groups and the cellular volume of classifying spaces as one passes to normal subgroups $G_n

Two groups are said to have the same nilpotent genus if they have the same nilpotent quotients. We answer four questions of Baumslag concerning nilpotent completions. (i) There exists a pair of finitely generated, residually torsion-free-nilpotent groups of the same nilpotent genus such that one is finitely presented and the other is not. (ii) There exists a pair of finitely presented, residually torsion-free-nilpotent groups of the same nilpotent genus such that one has a solvable conjugacy problem and the other does not. (iii) There exists a pair of finitely generated, residually torsion-free-nilpotent groups of the same nilpotent genus such that one has finitely generated second homology $H_2(-,\Z)$ and the other does not. (iv) A non-trivial normal subgroup of infinite index in a finitely generated parafree group cannot be finitely generated. In proving this last result, we establish that the first $L^2$ betti number of a finitely generated parafree group of rank $r$ is $r-1$. It follows that the reduced $C^*$-algebra of the group is simple if $r\ge 2$, and that a version of the Freiheitssatz holds for parafree groups., Comment: Version accepted for Int Math Res Notices

We establish {\em{virtual surjection to pairs}} (VSP) as a general criterion for the finite presentability of subdirect products of groups: if $\Gamma_1,...,\Gamma_n$ are finitely presented and $S<\Gamma_1\times...\times\Gamma_n$ projects to a subgroup of finite index in each $\Gamma_i\times\Gamma_j$, then $S$ is finitely presentable, indeed there is an algorithm that will construct a finite presentation for $S$. We use the VSP criterion to characterise the finitely presented residually free groups. We prove that the class of such groups is recursively enumerable. We describe an algorithm that, given a finite presentation of a residually free group, constructs a canonical embedding into a direct product of finitely many limit groups. We solve the (multiple) conjugacy problem and membership problem for finitely presentable subgroups of residually free groups. We also prove that there is an algorithm that, given a finite generating set for such a subgroup, will construct a finite presentation. New families of subdirect products of free groups are constructed, including the first examples of finitely presented subgroups that are neither ${\rm{FP}}_\infty$ nor of Stallings-Bieri type, Comment: 44 pages. To appear in American Journal of Mathematics. This is a substantial rewrite of our previous Arxiv article 0809.3704, taking into account subsequent developments, advice of colleagues and referee's comments

There exist right angled Artin groups $A$ such that the isomorphism problem for finitely presented subgroups of $A$ is unsolvable, and for certain finitely presented subgroups the conjugacy and membership problems are unsolvable. It follows that if $S$ is a surface of finite type and the genus of $S$ is sufficiently large, then the corresponding decision problems for the mapping class group $Mod(S)$ are unsolvable. Every virtually special group embeds in the mapping class group of infinitely many closed surfaces. Examples are given of finitely presented subgroups of mapping class groups that have infinitely many conjugacy classes of torsion elements., Comment: 10 pages, no figures

We consider actions of automorphism groups of free groups by semisimple isometries on complete CAT$(0)$ spaces. If $n\ge 4$ then each of the Nielsen generators of Aut$(F_n)$ has a fixed point. If $n=3$ then either each of the Nielsen generators has a fixed point, or else they are hyperbolic and each Nielsen-generated $\Z^4\subset Aut(F_3)$ leaves invariant an isometrically embedded copy of Euclidean 3-space on which it acts as a discrete group of translations with the rhombic dodecahedron as a fundamental domain. An abundance of actions of the second kind is described. Constraints on maps from Aut$(F_n)$ to mapping class groups and linear groups are obtained. If $n\ge 2$ then neither Aut$(F_n)$ nor Out$(F_n)$ is the fundamental group of a compact K\"ahler manifold., Comment: 14 pages, no figures

We show that the isomorphism problem is solvable in the class of central extensions of word-hyperbolic groups, and that the isomorphism problem for biautomatic groups reduces to that for biautomatic groups with finite centre. We describe an algorithm that, given an arbitrary finite presentation of an automatic group $\Gamma$, will construct explicit finite models for the skeleta of $K(\Gamma,1)$ and hence compute the integral homology and cohomology of $\Gamma$., Comment: 21 pages, 4 figures

A group $\Gamma$ is defined to be cofinitely Hopfian if every homomorphism $\Gamma\to\Gamma$ whose image is of finite index is an automorphism. Geometrically significant groups enjoying this property include certain relatively hyperbolic groups and many lattices. A knot group is cofinitely Hopfian if and only if the knot is not a torus knot. A free-by-cyclic group is cofinitely Hopfian if and only if it has trivial centre. Applications to the theory of open mappings between manifolds are presented., Comment: 14 pages

For n > 2, the Dehn functions of Aut(F_n) and Out(F_n) are exponential. Hatcher and Vogtmann proved that they are at most exponential, and the complementary lower bound in the case n=3 was established by Bridson and Vogtmann. Handel and Mosher completed the proof by reducing the lower bound for n>4 to the case n=3. In this note we give a shorter, more direct proof of this last reduction., Comment: Final version, to appear in Annales de l'Institut Fourier

We explore the existence of homomorphisms between outer automorphism groups of free groups Out(F_n) \to Out(F_m). We prove that if n > 8 is even and n \neq m \leq 2n, or n is odd and n \neq m \leq 2n - 2, then all such homomorphisms have finite image; in fact they factor through det: Out(F_n) \to Z/2. In contrast, if m = r^n(n - 1) + 1 with r coprime to (n - 1), then there exists an embedding Out(F_n) \to Out(F_m). In order to prove this last statement, we determine when the action of Out(F_n) by homotopy equivalences on a graph of genus n can be lifted to an action on a normal covering with abelian Galois group., Comment: Final version, to appear in Mathematische Annalen. Minor errors and typos corrected, including range of n in Theorem B

If $G$ is a semisimple Lie group of real rank at least 2 and $\Gamma$ is an irreducible lattice in $G$, then every homomorphism from $\Gamma$ to the outer automorphism group of a finitely generated free group has finite image., Comment: 11 pages, no figures. Final version. To appear in Compositio Math

We exhibit classes of groups in which the word problem is uniformly solvable but in which there is no algorithm that can compute finite presentations for finitely presentable subgroups. Direct products of hyperbolic groups, groups of integer matrices, and right-angled Coxeter groups form such classes. We discuss related classes of groups in which there does exist an algorithm to compute finite presentations for finitely presentable subgroups. We also construct a finitely presented group that has a polynomial Dehn function but in which there is no algorithm to compute the first Betti number of the finitely presentable subgroups., Comment: Final version. To appear in GGD volume dedicated to Fritz Grunewald

Whenever the mapping class group of a closed orientable surface of genus g acts by semisimple isometries on a complete CAT(0) space of dimension less than g it fixes a point., Comment: 14 pages, 1 figure. Minor edits following referee's comments. To appear in Crelle's journal

Let S be an orientable surface of finite type and let Mod(S) be its mapping class group. We consider actions of Mod(S) by semisimple isometries on complete CAT(0) spaces. If the genus of S is at least 3, then in any such action all Dehn twists act as elliptic isometries. The action of Mod(S) on the completion of Teichm\"uller space with the Weil-Petersson metric shows that there are interesting actions of this type. Whenever the mapping class group of a closed orientable surface of genus g acts by semisimple isometries on a complete CAT(0) space of dimension less than g it must fix a point. The mapping class group of a closed surface of genus 2 acts properly by semisimple isometries on a complete CAT(0) space of dimension 18., Comment: To appear in "The Geometry of Riemann Surfaces", LMS Lecture Notes 368. Dedicated to Bill Harvey on his 65th birthday. 12 pages no figures