Searcharxiv⌕ Search

arXiv subjects

Pallavi Dani

Publications and source records attributed to Pallavi Dani.

At least 19 recordsLinked to original sources
arXiv2025

Conformal dimension bounds, Pontryagin sphere boundaries, and algebraic fibering of right-angled Coxeter groups

We introduce a graph-theoretic condition, called $(n,m)$--branching, that ensures a combinatorial round tree with controlled branching parameters can be quasi-isometrically embedded in the Davis complex of the right-angled Coxeter group defined by the graph. This construction yields a lower bound on the conformal dimension of the boundary of such a hyperbolic group. We exhibit numerous families of graphs with this property, including many 1-dimensional spherical buildings. We prove an embedding result, showing that under mild hypotheses a flag-no-square graph embeds as an induced subgraph in a flag-no-square triangulation of a closed surface. We use this to embed our branching graphs into graphs presenting hyperbolic right-angled Coxeter groups with Pontryagin sphere boundary. We conclude there are examples of such groups with conformal dimension tending to infinity, and hence, there are infinitely many quasi-isometry classes within this family. We use conformal dimension to show that recent work of Lafont--Minemyer--Sorcar--Stover--Wells can be upgraded to conclude that for every $n \geq 2$ there exist infinitely many quasi-isometry classes of hyperbolic right-angled Coxeter groups that virtually algebraically fiber and have virtual cohomological dimension $n$.

math.GR↗
arXiv2025

RAAGedy right-angled Coxeter groups

We give criteria for deciding whether or not a triangle-free simple graph is the presentation graph of a right-angled Coxeter group that is quasiisometric to some right-angled Artin group, and, if so, producing a presentation graph for such a right-angled Artin group. We introduce two new graph modification operations, cloning and unfolding, to go along with an existing operation called link doubling. These operations change the presentation graph but not the quasiisometry type of the resulting group. We give criteria on the graph that imply it can be transformed by these operations into a graph that is recognizable as presenting a right-angled Coxeter group commensurable to a right-angled Artin group. In the converse direction we derive coarse geometric obstructions to being quasiisometric to a right-angled Artin group, first by specializing existing results from the literature to this setting, then by developing new approaches using configurations of maximal product regions. In all cases we give sufficient graphical conditions that imply these geometric obstructions. We implemented our criteria on a computer and applied them to an enumeration of small graphs. Our methods completely answer the motiving question when the graph has at most 10 vertices.

math.GR↗
arXiv2024

Fractional distortion in hyperbolic groups

For all integers $p>q>0$ and $k >0$, and all non-elementary torsion-free hyperbolic groups $H$, we construct a hyperbolic group $G$ in which $H$ is a subgroup, such that the distortion function of $H$ in $G$ grows like $\exp^k(n^{p/q})$. Here, $\exp^k$ denotes the $k$-fold-iterated exponential function.

math.GR↗
arXiv2022

Divergence, thickness and hypergraph index for general Coxeter groups

We study divergence and thickness for general Coxeter groups $W$. We first characterise linear divergence, and show that if $W$ has superlinear divergence then its divergence is at least quadratic. We then formulate a computable combinatorial invariant, hypergraph index, for arbitrary Coxeter systems $(W,S)$. This generalises Levcovitz's definition for the right-angled case. We prove that if $(W,S)$ has finite hypergraph index $h$, then $W$ is (strongly algebraically) thick of order at most $h$, hence has divergence bounded above by a polynomial of degree $h+1$. We conjecture that these upper bounds on the order of thickness and divergence are in fact equalities, and we prove our conjecture for certain families of Coxeter groups. These families are obtained by a new construction which, given any right-angled Coxeter group, produces infinitely many examples of non-right-angled Coxeter systems with the same hypergraph index. Finally, we give an upper bound on the hypergraph index of any Coxeter system $(W,S)$, and hence on the divergence of $W$, in terms of, unexpectedly, the topology of its associated Dynkin diagram.

math.GR↗
arXiv2022

Right-angled Artin subgroups of right-angled Coxeter and Artin groups

We determine when certain natural classes of subgroups of right-angled Coxeter groups (RACGs) and right-angled Artin groups (RAAGs) are themselves RAAGs. We characterize finite-index visual RAAG subgroups of 2-dimensional RACGs. As an application, we show that any 2-dimensional, one-ended RACG with planar defining graph is quasi-isometric to a RAAG if and only if it is commensurable to a RAAG. Additionally, we give new examples of RACGs with non-planar defining graphs which are commensurable to RAAGs. Finally, we give a new proof of a result of Dyer: every subgroup generated by conjugates of RAAG generators is itself a RAAG.

math.GR↗
arXiv2022

Non-quasiconvex subgroups of hyperbolic groups via Stallings-like techniques

We provide a new method of constructing non-quasiconvex subgroups of hyperbolic groups by utilizing techniques inspired by Stallings' foldings. The hyperbolic groups constructed are in the natural class of right-angled Coxeter groups (RACGs for short) and can be chosen to be 2-dimensional. More specifically, given a non-quasiconvex subgroup of a (possibly non-hyperbolic) RACG, our construction gives a corresponding non-quasiconvex subgroup of a hyperbolic RACG. We use this to construct explicit examples of non-quasiconvex subgroups of hyperbolic RACGs including subgroups whose generators are as short as possible (length two words), finitely generated free subgroups, non-finitely presentable subgroups, and subgroups of fundamental groups of square complexes of nonpositive sectional curvature.

math.GR↗
arXiv2021

Subgroups of right-angled Coxeter groups via Stallings-like techniques

We associate cube complexes called completions to each subgroup of a right-angled Coxeter group (RACG). A completion characterizes many properties of the subgroup such as whether it is quasiconvex, normal, finite-index or torsion-free. We use completions to show that reflection subgroups are quasiconvex, as are one-ended Coxeter subgroups of a 2-dimensional RACG. We provide an algorithm that determines whether a given one-ended, 2-dimensional RACG is isomorphic to some finite-index subgroup of another given RACG. In addition, we answer several algorithmic questions regarding quasiconvex subgroups. Finally, we give a new proof of Haglund's result that quasiconvex subgroups of RACGs are separable.

math.GT↗
arXiv2019

Right-angled Coxeter groups with non-planar boundary

We investigate the planarity of the boundaries of right-angled Coxeter groups. We show that non-planarity of the defining graph does not necessarily imply non-planarity of every boundary of the associated right-angled Coxeter group, although it does in many cases. Our techniques yield a characterization of the triangle-free defining graphs such that the associated right-angled Coxeter group has boundary a Menger curve.

math.GT↗
arXiv2018

The large-scale geometry of right-angled Coxeter groups

This is a survey of some aspects of the large-scale geometry of right-angled Coxeter groups. The emphasis is on recent results on their negative curvature properties, boundaries, and their quasi-isometry and commensurability classification.

math.GR↗
arXiv2017

Bowditch's JSJ tree and the quasi-isometry classification of certain Coxeter groups, with an appendix written jointly with Christopher Cashen

Bowditch's JSJ tree for splittings over 2-ended subgroups is a quasi-isometry invariant for 1-ended hyperbolic groups which are not cocompact Fuchsian. Our main result gives an explicit, computable "visual" construction of this tree for certain hyperbolic right-angled Coxeter groups. As an application of our construction we identify a large class of such groups for which the JSJ tree, and hence the visual boundary, is a complete quasi-isometry invariant, and thus the quasi-isometry problem is decidable. We also give a direct proof of the fact that among the Coxeter groups we consider, the cocompact Fuchsian groups form a rigid quasi-isometry class. In an appendix, written jointly with Christopher Cashen, we show that the JSJ tree is not a complete quasi-isometry invariant for the entire class of Coxeter groups we consider.

math.GR↗
arXiv2017

Commensurability for certain right-angled Coxeter groups and geometric amalgams of free groups

We give explicit necessary and sufficient conditions for the abstract commensurability of certain families of 1-ended, hyperbolic groups, namely right-angled Coxeter groups defined by generalized theta-graphs and cycles of generalized theta-graphs, and geometric amalgams of free groups whose JSJ graphs are trees of diameter at most 4. We also show that if a geometric amalgam of free groups has JSJ graph a tree, then it is commensurable to a right-angled Coxeter group, and give an example of a geometric amalgam of free groups which is not quasi-isometric (hence not commensurable) to any group which is finitely generated by torsion elements. Our proofs involve a new geometric realization of the right-angled Coxeter groups we consider, such that covers corresponding to torsion-free, finite-index subgroups are surface amalgams.

math.GR↗
arXiv2013

Divergence in right-angled Coxeter groups

Let W be a 2-dimensional right-angled Coxeter group. We characterise such W with linear and quadratic divergence, and construct right-angled Coxeter groups with divergence polynomial of arbitrary degree. Our proofs use the structure of walls in the Davis complex.

math.GR↗
arXiv2012

Homological and homotopical Dehn functions are different

The homological and homotopical Dehn functions are different ways of measuring the difficulty of filling a closed curve inside a group or a space. The homological Dehn function measures fillings of cycles by chains, while the homotopical Dehn function measures fillings of curves by disks. Since the two definitions involve different sorts of boundaries and fillings, there is no a priori relationship between the two functions, but prior to this work there were no known examples of finitely-presented groups for which the two functions differ. This paper gives the first such examples, constructed by amalgamating a free-by-cyclic group with several Bestvina-Brady groups.

math.GR↗
arXiv2011

Pushing fillings in right-angled Artin groups

We construct "pushing maps" on the cube complexes that model right-angled Artin groups (RAAGs) in order to study filling problems in certain subsets of these cube complexes. We use radial pushing to obtain upper bounds on higher divergence functions, finding that the k-dimensional divergence of a RAAG is bounded by r^{2k+2}. These divergence functions, previously defined for Hadamard manifolds to measure isoperimetric properties "at infinity," are defined here as a family of quasi-isometry invariants of groups; thus, these results give new information about the QI classification of RAAGs. By pushing along the height gradient, we also show that the k-th order Dehn function of a Bestvina-Brady group is bounded by V^{(2k+2)/k}. We construct a class of RAAGs called "orthoplex groups" which show that each of these upper bounds is sharp.

math.GR↗
arXiv2009

Morse theory and conjugacy classes of finite subgroups II

We construct a hyperbolic group with a finitely presented subgroup, which has infinitely many conjugacy classes of finite-order elements. We also use a version of Morse theory with high dimensional horizontal cells and use handle cancellation arguments to produce other examples of subgroups of CAT(0) groups with infinitely many conjugacy classes of finite-order elements.

math.GR↗
arXiv2007

Morse theory and conjugacy classes of finite subgroups

We construct a CAT(0) group containing a finitely presented subgroup with infinitely many conjugacy classes of finite-order elements. Unlike previous examples (which were based on right-angled Artin groups) our ambient CAT(0) group does not contain any rank 3 free abelian subgroups. We also construct examples of groups of type F_n inside mapping class groups, Aut(F), and Out(F) which have infinitely many conjugacy classes of finite-order elements.

math.GR↗