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In this paper it is proved that the group $F\left(\frac32\right)$, a Thompson-style group with breaks in $\mathbb{Z}\left[\frac16\right]$ but whose slopes are restricted only to powers of $\frac32$, is finitely generated, with a generating set of two elements.

We relaxe the constraint on the domains of combinatorial HHS machinery so combinatorial HHS machinery works for most cubical curve graphs. As an application we extend the factor system machinery of the CAT(0) cube complex to the quasi-median graphs., Comment: 34 pages

We show that the Morse boundary of a Morse local-to-global group is $\sigma$-compact. Moreover, we show that the converse holds for small cancellation groups. As an application, we show that the Morse boundary of a non-hyperbolic, Morse local-to-global group that has contraction does not admit a non-trivial stationary measure. In fact, we show that any stationary measure on a geodesic boundary of such a groups needs to assign measure zero to the Morse boundary. Unlike previous results, we do not need any assumptions on the stationary measures considered., Comment: 36 pages, 9 figures. Comments are welcome!

We prove that higher-dimensional Thompson's groups have linear divergence functions. By the work of Dru\c{t}u, Mozes, and Sapir, this implies none of the asymptotic cones of $nV$ has a cut-point., Comment: 19 pages, 11 figures

A group has Property $\mathrm{(NL)}$ if it does not admit a loxodromic element in any hyperbolic action. In other words, a group with this property is inaccessible for study from the perspective of hyperbolic actions. This property was introduced by Balasubramanya, Fournier-Facio and Genevois, who initiated the study of this property. We expand on this research by studying Property $\mathrm{(NL)}$ in Coxeter groups, a class of groups that are defined by an underlying graph. One of our main results show that a right-angled Coxeter group (RACG) has Property $\mathrm{(NL)}$ if and only if its defining graph is complete. We then move beyond the right-angled case to show that if a defining graph is disconnected, its corresponding Coxeter group does not have Property $\mathrm{(NL)}$. Lastly, we classify which triangle groups (Coxeter groups with three generators) have Property $\mathrm{(NL)}$., Comment: 11 pages, 6 figures

Given a geodesic metric space $X$, we construct a corresponding hyperbolic space, which we call the contraction space, that detects all strongly contracting directions in the following sense; a geodesic in $X$ is strongly contracting if and only if its parametrized image in the contraction space is a quasi-geodesic. If a finitely generated group $G$ acts geometrically on $X$, then all strongly-contracting elements act as WPD elements on the contraction space. If the space $X$ is CAT(0), or more generally Morse-dichotomous, that is if all Morse geodesics are strongly-contracting, then all generalized loxodromics act as WPD elements, implying that the action is what we call ``universally WPD''., Comment: 27 pages, 5 figures. Comments welcome!

In 2018, Kielak gave a virtual fibering criterion for RFRS groups. In this paper, we present a simpler proof of this., Comment: 14 pages, added a characterization of finitely generated RFRS groups due to Sam Fisher and Dawid Kielak, other small changes

In this paper we explore the interplay between aspects of the geometry and algebra of three families of groups of the form B semidirect the integers Z, namely Lamplighter groups, solvable Baumslag-Solitar groups and lattices in SOL. In particular we examine what kind of maps are induced on B by quasi-isometries that coarsely permute cosets of the Z subgroup. By the results of Schwartz(1996) and Taback(2000) in the lattice in SOL and solvable Baumslag-Solitar cases respectively such quasi-isometries induce affine maps of B. We show that this is no longer true in the lamplighter case but the induced maps do share some features with affine maps., Comment: 20 pages, 6 figures

We study finitely generated pairs of groups $H \leq G$ such that the Schreier graph of $H$ has at least two ends and is \emph{narrow}. Examples of narrow Schreier graphs include those that are quasi-isometric to finitely ended trees or have linear growth. Under this hypothesis, we show that $H$ is a virtual fiber subgroup if and only if $G$ contains infinitely many double cosets of $H$. Along the way, we prove that if a group acts essentially on a finite dimensional CAT(0) cube complex with no facing triples then it virtually surjects onto the integers with kernel commensurable to a hyperplane stabiliser., Comment: 17 pages, 2 figures. V2 extends the main result to a wider class of subgroups, V3 implements the referee's comments. To appear in Bull. Lond. Math. Soc

We show that under appropriate assumptions, a blown-up corona of a relatively hyperbolic group is equivariant and the compactification of the universal space for proper action by the blown-up corona is contractible. As a corollary, we establish the formula to determine the covering dimension of the blown-up corona by the cohomological dimension of the group. We also show that the blown-up corona of a hyperbolic group with respect to an almost malnormal family of quasiconvex subgroups is homeomorphic to the Gromov boundary of the group., Comment: 32 pages

This paper presents a study of the well-known marked length spectrum rigidity problem in the coarse-geometric setting. For any two (possibly non-proper) group actions $G\curvearrowright X_1$ and $G\curvearrowright X_2$ with contracting property, we prove that if the two actions have the same marked length spectrum, then the orbit map $Go_1\to Go_2$ must be a rough isometry. In the special case of cusp-uniform actions, the rough isometry can be extended to the entire space. This generalizes the existing results in hyperbolic groups and relatively hyperbolic groups. In addition, we prove a finer marked length spectrum rigidity from confined subgroups and further, geometrically dense subgroups. Our proof is based on the Extension Lemma and uses purely elementary metric geometry. This study produces new results and recovers existing ones for many more interesting groups through a unified and elementary approach., Comment: 45 pages; updated information about reference [41]

In this paper, we study the asymptotics of several growth functions related to twisted conjugacy on virtually abelian groups. First, we study the twisted conjugacy growth function, which counts the number of twisted conjugacy classes intersecting the ball of radius r around the identity element. Thereafter we study the function that measures the size of the intersection of a given twisted conjugacy class with the balls around the identity element. Finally, we study the number of induced twisted conjugacy classes in certain finite quotients of the given virtually abelian group. In each of these cases we obtain a polynomial asymptotic behaviour of these growth functions.

We construct explicit finite generating sets for the stabilizers in Thompson's group $F$ of rational points of a unit interval or a Cantor set. Our technique is based on the Reidemeister-Schreier procedure in the context of Schreier graphs of such stabilizers in $F$. It is well known that the stabilizers of dyadic rational points are isomorphic to $F\times F$ and can thus be generated by 4 explicit elements. We show that the stabilizer of every non-dyadic rational point $b\in (0,1)$ is generated by 5 elements that are explicitly calculated as words in generators $x_0, x_1$ of $F$ that depend on the binary expansion of $b$. We also provide an alternative simple proof that the stabilizers of all rational points are finitely presented., Comment: 19 pages, 9 figures and pictures

We outline a general procedure that builds classifying spaces for generalized Thompson groups $\Gamma$. The construction depends on a small number of choices: (1) an inverse semigroup $S$ of partial transformations that ``locally determine" $\Gamma$; (2) an equivalence relation on certain pairs $(f,D)$, and (3) an ``expansion" rule $\mathcal{E}$. These choices determine an \emph{expansion set} $\mathcal{B}$, which is a combinatorial device that outputs a simplicial complex $\Delta^{f}_{\mathcal{B}}$ upon which $\Gamma$ acts. Under favorable conditions, often achieved in practice, $\Delta^{f}_{\mathcal{B}}$ is contractible, and the action of $\Gamma$ has small stabilizers. The definition of $\Delta^{f}_{\mathcal{B}}$ is such that ascending and descending links in $\Delta^{f}_{\mathcal{B}}$ can be described via formulas that depend only on the expansion rule $\mathcal{E}$. The result is to facilitate the usual computations of the connectivity of the descending link. Under natural hypotheses, one can prove that the acting group has type $F_{\infty}$. The net effect of our results is to automate results of this kind. Several applications are given; in particular, we sketch unified proofs that $V$, $nV$, R\"{o}ver's group $G$, and the Lodha-Moore group have type $F_{\infty}$., Comment: 50 pages, 13 figures (The current version corrects Remark 4.12 and adds an acknowledgement at the end of the introduction. Otherwise, the paper is unchanged.)

We use Gersten's generalization of Whitehead's algorithm to determine whether a given finitely generated subgroup of a free group $F$ is elliptic in an elementary cyclic splitting of $F$. We provide a similar result for all elementary cyclic splittings of a free group of rank two and partial results towards an algorithm for ellipticity in elementary cyclic splittings of $F$ as an HNN-extension., Comment: 24 pages, 7 figures

This paper studies the locally uniform exponential growth and product set growth for a finitely generated group $G$ acting properly on a finite product of hyperbolic spaces. Under the assumption of coarsely dense orbits or shadowing property on factors, we prove that any finitely generated non-virtually abelian subgroup has uniform exponential growth. These assumptions are fulfilled in many hierarchically hyperbolic groups, including mapping class groups, specially cubulated groups and BMW groups. Moreover, if $G$ acts weakly acylindrically on each factor, we show that, with two exceptional classes of subgroups, $G$ has uniform product set growth. As corollaries, this gives a complete classification of subgroups with product set growth for any group acting discretely on a simply connected manifold with pinched negative curvature, for groups acting acylindrically on trees, and for 3-manifold groups., Comment: 32 pages, 4 figures. Corrections and supplements have been made in response to referee comments

We show that for any finitely generated subgroup $H$ of a limit group $L$ there exists a finite-index subgroup $K$ containing $H$, such that $K$ is a subgroup of a group obtained from $H$ by a series of extensions of centralizers and free products with $\mathbb Z$. If $H$ is non-abelian, the $K$ is fully residually $H$. We also show that for any finitely generated subgroup of a limit group, there is a finite-dimensional representation of the limit group which separates the subgroup in the induced Zariski topology. As a corollary, we establish a polynomial upper bound on the size of the quotients used to separate a finitely generated subgroup in a limit group. This generalizes the results of Louder, McReynolds and Patel. Another corollary is that a hyperbolic limit group satisfies the Geometric Hanna Neumann conjecture.

We show that the sublinearly Morse boundary of a CAT(0) cubical group with a factor system is well-defined up to homeomorphism with respect to the visual topology. The key tool used in the proof is a new topology on sublinearly Morse boundaries that is induced by group actions on hyperbolic spaces that are sufficiently nice, for example, largest acylindrical actions. Using the same techniques, we obtain a explicit description of this new topology on the sublinearly Morse boundary of any hierarchically hyperbolic group in terms of medians. Finally, we explicitly describe the sublinear Morse boundaries of graph manifolds using their actions on Bass-Serre trees., Comment: 49 pages, 3 figures; comments welcome

A sublinear biLipschitz equivalence (SBE) between metric spaces is a map from one space to another that distorts distances with bounded multiplicative constants and sublinear additive error. Given any sublinear function $\kappa$, $\kappa$-Morse boundaries are defined for all geodesic proper metric spaces as a quasi-isometrically invariant and metrizable topological space of quasi-geodesic rays. In this paper, we prove that $\kappa$-Morse boundaries of proper geodesic metric spaces are invariant under suitable SBEs. A tool in the proof is the use of sublinear rays, that is, sublinear bilispchitz embeddings of the half line, generalizing quasi-geodesic rays. As an application we distinguish a pair of right-angled Coxeter groups brought up by Behrstock up to sublinear biLipschitz equivalence. We also show that under mild assumptions, generic random walks on countable groups are sublinear rays., Comment: 22 pages, 8 figures

We introduce the notion of quasi-BNS invariants, where we replace homomorphism to $\mathbb R$ by homogenous quasimorphisms to $\mathbb R$ in the theory of Bieri-Neumann-Strebel invariants. We prove that the quasi-BNS invariant $Q\Sigma(G)$ of a finitely generated group $G$ is open; we connect it to approximate finite generation of almost kernels of homogenous quasimorphisms; finally we prove a Sikorav-style theorem connecting $Q\Sigma(G)$ to the vanishing of the suitably defined Novikov homology., Comment: 13 pages, no figures

Motivated by the work of Bestvina-Feighn ([BF92]) and Mj-Sardar ([MS12]), we define trees of metric bundles subsuming both the trees of metric spaces and the metric bundles. Then we prove a combination theorem for these spaces. More precisely, we prove that a tree of metric bundles is hyperbolic if the following hold (see Theorem 1.4). $(1)$ The fibers are uniformly hyperbolic metric spaces and the base is also hyperbolic metric space, $(2)$ barycenter maps for the fibers are uniformly coarsely surjective, $(3)$ the edge spaces are uniformly qi embedded in the corresponding fibers and $(4)$ Bestvina-Feighn's flaring condition is satisfied. As an application, we provide a combination theorem for certain complexes of groups over finite simplicial complex (see Theorem 1.2)., Comment: Title and abstract have been changed, Section 4 is added and accordingly, sections have been rearranged, more figures are added; in this version, we work with only spaces not with approximating graph and subsequently, some of the statements and their proofs have been modified, 8 figures, 63 pages. Comments and Suggestions are welcome

Let $G_{n,k}$ denote the real Grassmann manifold of $k$-dimensional vector subspaces of $\mathbb R^n$. Using the Hodgkin spectral sequence, we compute the complex $K$-ring of $G_{n,k}$, up to a small indeterminacy, for all values of $n,k$ where $2\le k\le n-2$. When $n\equiv 0\!\!\mod 4, k\equiv 1\!\!\mod 2$ our result is complete., Comment: 21 pages, Minor typographical changes have been made. To appear on Homology, Homotopy and Applications

We generalized the Lodha-Moore group into $n$-adic and showed analogues in the Lodha-Moore group of properties between the Thompson group $F$ and the generalized Thompson group $F(n)$., Comment: 27 pages, 6 figures

Given a metric (graph) bundle $X$ over $B$ where all the fibres are strongly relatively hyperbolic and nonelementary we show that, under certain conditions, $X$ is strongly hyperbolic relative to a collection of maximal cone-subbundles of horosphere-like spaces. Further, given a coarsely Lipschitz qi embedding $i: A\to B$, we show that the pullback $Y$ is strongly relatively hyperbolic and the map $Y\to X$ admits a Cannon-Thurston (CT) map. As an application, we prove a group-theoretic analogue of this result for a relatively hyperbolic extension of groups., Comment: 40 pages, 4 figures

We show there is a finite-volume, hyperbolic $7$-manifold that algebraically fibres with finitely presented kernel of type $\mathtt{FP}(\mathbb Q)$. This manifold is a finite cover of the one constructed by Italiano--Martelli--Migliorini., Comment: Lemma 3.2 of the previous version is strengthened (now Proposition 3.3). Accepted for publication in the Bulletin of the LMS

We show that if G is an admissible group acting geometrically on a CAT(0) space X, then G is a hierarchically hyperbolic space and with mild assumptions the sublinearly-Morse boundary of the group is a topological model for associated Poisson boundaries ., Comment: 27 pages, 3 figures

Given a finitely generated subgroup $H$ of a finitely generated group $G$ and a non-principal ultrafilter $\omega$, we consider a natural subspace, $Cone^{\omega}_{G}(H)$, of the asymptotic cone of $G$ corresponding to $H$. Informally, this subspace consists of the points of the asymptotic cone of $G$ represented by elements of the ultrapower $H^{\omega}$. We show that the connectedness and convexity of $Cone^{\omega}_{G}(H)$ detect natural properties of the embedding of $H$ in $G$. We begin by defining a generalization of the distortion function and show that this function determines whether $Cone^{\omega}_{G}(H)$ is connected. We then show that whether $H$ is strongly quasi-convex in $G$ is detected by a natural convexity property of $Cone^{\omega}_{G}(H)$ in the asymptotic cone of $G$., Comment: 20 pages, 4 figures, added a citation to a paper of Behrstock

We show that a virtually RFRS group $G$ of type $\mathrm{FP}_n(\mathbb{Q})$ virtually algebraically fibres with kernel of type $\mathrm{FP}_n(\mathbb{Q})$ if and only if the first $n$ $\ell^2$-Betti numbers of $G$ vanish, that is, $b_p^{(2)}(G) = 0$ for $0 \leqslant p \leqslant n$. We also offer a variant of this result over other fields, in particular in positive characteristic. As an application of the main result, we show that virtually amenable RFRS groups of type $\mathrm{FP}(\mathbb{Q})$ are polycyclic-by-finite. It then follows that if $G$ is a virtually RFRS group of type $\mathrm{FP}(\mathbb{Q})$ such that $\mathbb{Z}G$ is Noetherian, then $G$ is polycyclic-by-finite. This answers a longstanding conjecture of Baer for virtually RFRS groups of type $\mathrm{FP}(\mathbb{Q})$., Comment: Version accepted for publication at Compositio Mathematica. An addendum (which will appear in the author's thesis, not in the published version of this article) has been added, giving a simple proof that virtually locally indicable groups of finite cohomological dimension satisfy Baer's Conjecture. This was proven in the previous version under the stronger assumptions RFRS and finite type

We compute the Bieri-Neumann-Strebel invariants $\Sigma^1$ for the generalized solvable Baumslag-Solitar groups $\Gamma_n$ and their finite index subgroups. Using $\Sigma^1$, we show that certain finite index subgroups of $\Gamma_n$ cannot be isomorphic to $\Gamma_{k}$ for any $k$. In addition, we use the BNS-invariants to give a new proof of property $R_\infty$ for the groups $\Gamma_n$ and their finite index subgroups., Comment: 20 pages

We show that every group acting freely and vertex-transitively by isometries on a product of two regular trees of finite valence is boundary rigid. That means that every CAT(0) space that admits a geometric action of any such group has the visual boundary homeomorphic to a join of two copies of the Cantor set., Comment: 10 pages. Upon the revision of the paper, we learned that Margolis (arXiv:2207.04401) proved a much stronger result than our main theorem