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We provide an explicit computation of the topological $K$-theory groups $K_*(C_r^*(\mathbb{Z}^n\rtimes \mathbb{Z}/m))$ of semidirect products of the form $\mathbb{Z}^n\rtimes \mathbb{Z}Z/m$ with $m$ square-free. We want to highlight the fact that we are not impossing any conditions on the $\Z/m$-action on $\mathbb{Z}^n$. This generalizes previous computations of L\"uck-Davis and Langer-L\"uck., Comment: 10 pages. Comments are welcome

In \cite{Ha86} Harer explicitly constructed a spine for the decorated Teichm\"uller space of orientable surfaces with at least one puncture and negative Euler characteristic. In this paper we point out some instances where his computation of the dimension of this spine is off by $1$ and give the correct dimension., Comment: 10 pages, 4 figures

We provide an explicit computation of the cohomology groups (with untwisted coefficients) of semidirect products of the form $\mathbb{Z}^n\rtimes \mathbb{Z}/m$ with $m$ free of squares, by means of formulas that only depend on $n$, $m$ and the action of $\mathbb{Z}/m$ on $\mathbb{Z}^n$. We want to highlight the fact that we are not impossing any conditions on the $\mathbb{Z}/m$-action on $\mathbb{Z}^n$, and as far as we know our formulas are the first in the literature in this generality. This generalizes previous computations of L\"uck-Davis and Adem-Ge-Pan-Petrosyan. In order to show that our formulas are usable, we develop a concrete example of the form $\mathbb{Z}^5\rtimes \mathbb{Z}/6$ where its cohomology groups are described in full detail.

We prove that the virtually cyclic (geometric) dimension of the finite index congruence subgroup $\mathrm{IA}_N(3)$ of $\mathrm{Out}(F_N)$ is $2N-2$. From this we deduce the virtually cyclic dimension of $\mathrm{Out}(F_N)$ is finite. Along the way we prove L\"uck's property (C) holds for $\mathrm{Out}(F_N)$, we prove that the commensurator of a cyclic subgroup of $\mathrm{IA}_N(3)$ equals its centraliser, we give an $\mathrm{IA}_N(3)$ analogue of various exact sequences arising from reduction systems for mapping class groups, and give a near complete description of centralisers of infinite order elements in $\mathrm{IA}_3(3)$., Comment: 35 pages, comments welcome

For a topological group $G$ let $E_{\textsf{com}}(G)$ be the total space of the universal transitionally commutative principal $G$-bundle as defined by Adem--Cohen--Torres-Giese. So far this space has been most studied in the case of compact Lie groups; but in this paper we focus on the case of infinite discrete groups. For a discrete group $G$, the space $E_{\textsf{com}}(G)$ is homotopy equivalent to the geometric realization of the order complex of the poset of cosets of abelian subgroups of $G$. We show that for fundamental groups of closed orientable geometric $3$-manifolds, this space is always homotopy equivalent to a wedge of circles. On our way to prove this result we also establish some structural results on the homotopy type of $E_{\textsf{com}}(G)$., Comment: 36 pages, 4 figures, comments very welcome!

Let $\mathrm{Mod}(S)$ be the mapping class group of a compact connected orientable surface $S$, possibly with punctures and boundary components, with negative Euler characteristic. We prove that for any infinite virtually abelian subgroup $H$ of $\mathrm{Mod}(S)$, there is a subgroup $H'$ commensurable with $H$ such that the commensurator of $H$ equals the normalizer of $H'$. As a consequence we give, for each $n \geq 2$, an upper bound for the geometric dimension of $\mathrm{Mod}(S)$ for the family of abelian subgroups of rank bounded by $n$. These results generalize work by Juan-Pineda--Trujillo-Negrete and Nucinkis--Petrosyan for the virtually cyclic case., Comment: 19 pages, 1 figure

This survey studies pairs $(G,\mathcal{P})$ with $G$ a finitely generated group and $\mathcal{P}$ a (finite) collection of subgroups of $G$. We explore the notion of quasi-isometry of such pairs and the notion of a qi-characteristic collection of subgroups. Both notions are abstractions of phenomena that have appeared repeatedly in the work of several people within the quasi-isometric rigidity realm., Comment: 14 pages

For a finitely generated group $G$, let $H(G)$ denote Bowditch's taut loop length spectrum. We prove that if $G=(A\ast B) / \langle\!\langle \mathcal R \rangle\!\rangle $ is a $C'(1/12)$ small cancellation quotient of a the free product of finitely generated groups, then $H(G)$ is equivalent to $H(A) \cup H(B)$. We use this result together with bounds for cohomological and geometric dimensions, as well as Bowditch's construction of continuously many non-quasi-isometric $C'(1/6)$ small cancellation $2$-generated groups to obtain our main result: Let $\mathcal{G}$ denote the class of finitely generated groups. The following subclasses contain continuously many one-ended non-quasi-isometric groups: $\bullet\left\{G\in \mathcal{G} \colon \underline{\mathrm{cd}}(G) = 2 \text{ and } \underline{\mathrm{gd}}(G) = 3 \right\}$ $\bullet\left\{G\in \mathcal{G} \colon \underline{\underline{\mathrm{cd}}}(G) = 2 \text{ and } \underline{\underline{\mathrm{gd}}}(G) = 3 \right\}$ $\bullet\left\{G\in \mathcal{G} \colon \mathrm{cd}_{\mathbb{Q}}(G)=2 \text{ and } \mathrm{cd}_{\mathbb{Z}}(G)=3 \right\}$ On our way to proving the aforementioned results, we show that the classes defined above are closed under taking relatively finitely presented $C'(1/12)$ small cancellation quotients of free products, in particular, this produces new examples of groups exhibiting an Eilenberg-Ganea phenomenon for families. We also show that if there is a finitely presented counter-example to the Eilenberg-Ganea conjecture, then there are continuously many finitely generated one-ended non-quasi-isometric counter-examples., Comment: 21 pages

For a finitely generated group $G$ and collection of subgroups $\mathcal{P}$ we prove that the relative Dehn function of a pair $(G,\mathcal{P})$ is invariant under quasi-isometry of pairs. Along the way we show quasi-isometries of pairs preserve almost malnormality of the collection and fineness of the associated coned off Cayley graphs. We also prove that for a cocompact simply connected combinatorial $G$-$2$-complex $X$ with finite edge stabilisers, the combinatorial Dehn function is well-defined if and only if the $1$-skeleton of $X$ is fine. We also show that if $H$ is a hyperbolically embedded subgroup of a finitely presented group $G$, then the relative Dehn function of the pair $(G, H)$ is well-defined. In the appendix, it is shown that show that the Baumslag-Solitar group $\mathrm{BS}(k,l)$ has a well-defined Dehn function with respect to the cyclic subgroup generated by the stable letter if and only if neither $k$ divides $l$ nor $l$ divides $k$., Comment: Appendix by Ashot Minasyan

For a group $G$, let $\mathcal{F}_{n}$ be the family of all the subgroups of $G$ containing a subgroup isomorphic to $\mathbb{Z}^{r}$ for some $r=0,1,2,\dots ,n$ of finite index. Joecken, Lafont and S\'anchez Salda\~na computed the $\mathcal{F}_1$-dimension of 3-manifold groups. The goal of this article is to compute the $\mathcal{F}_n$-geometric dimension of 3-manifold groups for all $n\geq 2$., Comment: 22 pages, 1 table. Proofs of Theorem 3.1, Theorem 4.5 c), and Proposition 5.1 were expanded for the sake of clarity. Title was changed

Let $G$ and $H$ be quasi-isometric finitely generated groups and let $P\leq G$; is there a subgroup $Q$ (or a collection of subgroups) of $H$ whose left cosets coarsely reflect the geometry of the left cosets of $P$ in $G$? We explore sufficient conditions for a positive answer. The article consider pairs of the form $(G,\mathcal{P})$ where $G$ is a finitely generated group and $\mathcal{P}$ a finite collection of subgroups, there is a notion of quasi-isometry of pairs, and quasi-isometrically characteristic collection of subgroups. A subgroup is qi-characteristic if it belongs to a qi-characteristic collection. Distinct classes of qi-characteristic collections of subgroups have been studied in the literature on quasi-isometric rigidity, we list in the article some of them and provide other examples. The first part of the article proves: if $G$ and $H$ are finitely generated quasi-isometric groups and $\mathcal{P}$ is a qi-characteristic collection of subgroups of $G$, then there is a collection of subgroups $\mathcal{Q}$ of $H$ such that $ (G, \mathcal{P})$ and $(H, \mathcal{Q})$ are quasi-isometric pairs. The second part of the article studies the number of filtered ends $\tilde e (G, P)$ of a pair of groups, a notion introduced by Bowditch, and provides an application of our main result: if $G$ and $H$ are quasi-isometric groups and $P\leq G$ is qi-characterstic, then there is $Q\leq H$ such that $\tilde e (G, P) = \tilde e (H, Q)$., Comment: Version 3. 25 pages. Updated list of references. Version accepted for publication in AGT

We provide descriptions of the Whitehead groups, and the algebraic $K$-theory groups, of the fundamental group of a connected, oriented, closed $3$-manifold in terms of Whitehead groups of their finite subgroups and certain Nil-groups. The main tools we use are: the K-theoretic Farrell-Jones isomorphism conjecture, the construction of models for the universal space for the family of virtually cyclic subgroups in 3-manifold groups, and both the prime and JSJ-decompositions together with the well-known geometrization theorem., Comment: 21 pages

Let $\mathcal{N}_g$ be the mapping class group of a non-orientable closed surface. We prove that the proper cohomological dimension, the proper geometric dimension, and the virtual cohomological dimension of $\mathcal{N}_g$ are equal whenever $g\neq 4,5$. In particular, there exists a model for the classifying space of $\mathcal{N}_g$ for proper actions of dimension $\mathrm{vcd}(\mathcal{N}_g)=2g-5$. Similar results are obtained for the mapping class group of a non-orientable surface with boundaries and possibly punctures, and for the pure mapping class group of a non-orientable surface with punctures and without boundaries., Comment: 20 pages. Version accepted for publication in Homology, Homotopy and Applications

We prove that the Bredon cohomological dimension and the virtual cohomological dimension coincide for groups that admit a cocompact model for $\underline{E}G$ and satisfy properties (M) and (NM). Among the examples of groups satisfying these hypothesis are cocompact and arithmetic Fuchsian groups, one relator groups, the Hilbert modular group and $3$-manifold groups., Comment: Version accepted for publication in Glasgow Mathematical Journal. The title was changed, also some other minor corrections

We construct new examples of groups with cohomological dimension 2 and geometric dimension 3 with respect to the families of finite subgroups, virtually abelian groups of bounded rank, and the family of virtually poly-cyclic subgroups. Our main ingredients are the examples constructed by Brady-Leary-Nucinckis and Fluch-Leary, and Bass-Serre theory., Comment: Version accepted for publication in Proceedings of the AMS. Minor corrections

Let $G$ be a group and $\mathcal{F}$ be a family of subgroups closed under conjugation and subgroups. A model for the classifying space $E_{\mathcal{F}} G$ is a $G$-CW-complex $X$ such that every isotropy group belongs to $\mathcal{F}$, and for all $H\in \mathcal{F}$ the fixed point subspace $X^H$ is contractible. The group $G$ is of type $\mathcal{F}\text{-}\mathrm{F}_{n}$ if it admits a model for $E_\mathcal{F} G$ with $n$-skeleton with compact orbit space. The main result of the article provides is a characterization of $\mathcal{F}\text{-}\mathrm{F}_{n}$ analogue to Brown's criterion for $\mathrm{FP}_n$. As applications we provide criteria for this type of finiteness properties with respect to families to be preserved by finite extensions, a result that contrast with examples of Leary and Nucinkis. We also recover L\"uck's characterization of property $\underline{\mathrm{F}}_n$ in terms of the finiteness properties of the Weyl groups., Comment: Version accepted for publication in JPAA

Let G be the fundamental group of a connected, closed, orientable 3-manifold. We explicitly compute its virtually cyclic geometric dimension. Among the tools we use are the prime and JSJ decompositions of M, several push-out type constructions, as well as some Bredon cohomology computations.

Let $G$ be a group, let $H$ be a subgroup of $G$ and let $\Or(G)$ be the orbit category. In this paper we extend the definition of the relative group (co)homology theories of the pair $(G,H)$ defined by Adamson and Takasu to have coefficients in an $\Or(G)$-module. There is a canonical comparison homomorphism defined by Cisneros-Molina and Arciniega-Nev\'arez from Takasu's theory to Adamson's one. We give a necessary and sufficient condition on the subgroup $H$ for which the comparison homomorphism is an isomorphism for all coefficients. We also use the L\"uck-Wiermann construction to introduce a long exact sequence for Adamson (co)homology. Finally, we provide some examples of explicit computations for the comparison homomorphism., Comment: Shortened version. This version was accepted for publication in Quaestiones Mathematicae

In this paper we provide descriptions of the Whitehead groups with coefficients in a ring of the Hilbert modular group and its reduced version, as well as for the topological K-theory of $C^*$-algebras, after tensoring with $\mathbb{Q}$, by computing the source of the assembly maps in the Farrell-Jones and the Baum-Connes conjecture respectively. We also construct a model for the classifying space of the Hilbert modular group for the family of virtually cyclic subgroups.

Let $G$ be a word hyperbolic group. We prove that the algebraic $K$-theory groups of $\dbZ [G]$, $K_n(\dbZ[G])$, have finite rank for all $n\in \dbZ$. For a few classes of groups, we give explicit formulas for the ranks of the algebraic $K$-theory groups of their group rings., Comment: 9 pages, 1 figure

We prove the $K$ and $L$ theoretic versions of the Fibered Isomorphism Conjecture of F. T. Farrell and L. E. Jones for braid groups on a surface., Comment: To appear in OSU special year proceedings

We give formulas for the Whitehead groups and the rational $K$-theory groups of the (integer group ring of the) Hilbert modular group in terms of its maximal finite subgroups., Comment: 15 pages