arXiv subjects
Publications and source records attributed to Sam Shepherd.
Bass and Kulkarni proved that any pair of free uniform lattices in the automorphism group of a tree have conjugates that both lie inside a third uniform lattice (which is not necessarily free). We show that this does not generalise to trees with fins. The construction of our counter-example involves working with a certain generalisation of the universal groups of Burger and Mozes.
We characterise when a rank $n$ generalised Baumslag-Solitar group is CAT(0) and when it is biautomatic.
We prove that if a group $G$ admits a virtually special action on a CAT(0) cube complex, then any product of convex-cocompact subgroups of $G$ is separable. Previously, this was only known for products of three subgroups, or in the case where $G$ is hyperbolic, or in some other more technical cases with additional assumptions on the subgroups (plus these previous results assume that the action of $G$ is cocompact). We also provide an application to the action of a virtually special cubulated group on its contact graph (and to some other actions of cubulated groups on graphs).
A rank $n$ generalized Baumslag-Solitar group is a group that splits as a finite graph of groups such that all vertex and edge groups are isomorphic to $\mathbb{Z}^n$. In this paper we classify these groups in terms of their separability properties. Specifically, we determine when they are residually finite, subgroup separable and cyclic subgroup separable.
We introduce the notion of halfspaces associated to a group splitting, and investigate the relationship between the coarse geometry of the halfspaces and the coarse geometry of the group. Roughly speaking, the halfspaces of a group splitting are subgraphs of the Cayley graph obtained by pulling back the halfspaces of the Bass--Serre tree. Our first theorem shows that (under mild conditions) any splitting of a one-ended group can be upgraded to a splitting where all the halfspaces are one-ended. Our second theorem demonstrates that a one-ended group usually has a JSJ splitting where all the halfspaces are one-ended. And our third theorem states that if a one-ended finitely presented group $G$ admits a splitting such that some edge stabilizer has more than one end, but the halfspaces associated to the edge stabilizer are one-ended, then $H^2(G,\mathbb ZG)\ne \{0\}$; in particular $G$ is not simply connected at infinity and $G$ is not an $n$-dimensional duality group for $n\geq3$.
We introduce the notion of graphical discreteness to group theory. A finitely generated group is graphically discrete if whenever it acts geometrically on a locally finite graph, the automorphism group of the graph is compact-by-discrete. Notable examples include finitely generated nilpotent groups, most lattices in semisimple Lie groups, and irreducible non-geometric 3-manifold groups. We show graphs of groups with graphically discrete vertex groups frequently have strong rigidity properties. We prove free products of one-ended virtually torsion-free graphically discrete groups are action rigid within the class of virtually torsion-free groups. We also prove quasi-isometric rigidity for many hyperbolic graphs of groups whose vertex groups are closed hyperbolic manifold groups and whose edge groups are non-elementary quasi-convex subgroups. This includes the case of two hyperbolic 3-manifold groups amalgamated along a quasi-convex malnormal non-abelian free subgroup. We provide several additional examples of graphically discrete groups and illustrate this property is not a commensurability invariant.
Neural network potentials for kaolinite minerals have been fitted to data extracted from density functional theory calculation that were performed using the revPBE + D3 and revPBE + vdW functionals. These potentials have then been used to calculate static and dynamic properties of the mineral. We show that revPBE + vdW is better at reproducing the static properties. However, revPBE + D3 does a better job of reproducing the experimental IR spectrum. We also consider what happens to these properties when a fully-quantum treatment of the nuclei is employed. We find that nuclear quantum effects (NQEs) do not make a substantial difference to the static properties. However, when NQEs are included the dynamic properties of the material change substantially.
The primary method for showing that a given cubulated group is hierarchically hyperbolic is by constructing a factor system on the cube complex. In this paper we show that such a construction is not always possible, namely we construct a cubulated group for which the cube complex does not have a factor system. We also construct a cubulated group for which the induced action on the contact graph is not acylindrical.
We prove that all cubulated groups are semistable at infinity. In doing so we prove two further results about cubulations of groups. The first of these states that any one-ended cubulated group has a cubulation for which all halfspaces are one-ended. The second states that any cubulated group has a cubulation for which all quarterspaces are deep -- analogous to the fact that passing to the essential core of a given cubulation ensures that all halfspaces are deep.
Let $\Gamma$ be a graph product of finite groups, with finite underlying graph, and let $\Delta$ be the associated right-angled building. We prove that a uniform lattice $\Lambda$ in the cubical automorphism group Aut$(\Delta)$ is weakly commensurable to $\Gamma$ if and only if all convex subgroups of $\Lambda$ are separable. As a corollary, any two finite special cube complexes with universal cover $\Delta$ have a common finite cover. An important special case of our theorem is where $\Gamma$ is a right-angled Coxeter group and $\Delta$ is the associated Davis complex. We also obtain an analogous result for right-angled Artin groups. In addition, we deduce quasi-isometric rigidity for the group $\Gamma$ when $\Delta$ has the structure of a Fuchsian building.
Hydrogen bonds are of paramount importance in the chemistry of clays, mediating the interaction between the clay surface and water, and for some materials between separate layers. It is well-established that the accuracy of a computational model for clays depends on the level of theory at which the electronic structure is treated. However, for hydrogen-bonded systems the motion of light H nuclei on the electronic potential energy surface is often affected by quantum delocalization. Using path integral molecular dynamics, we show that nuclear quantum effects lead to a relatively small change in the structure of clays, but one that is comparable to the variation incurred by treating the clay at different levels of electronic structure theory. Accounting for quantum effects weakens the hydrogen bonds in clays, with H-bonds between different layers of the clay affected more than those within the same layer; this is ascribed to the fact that the confinement of an H atom inside a layer is independent of its participation in hydrogen-bonding. More importantly, the weakening of hydrogen bonds by nuclear quantum effects causes changes in the vibrational spectra of these systems, significantly shifting the O--H stretching peaks and meaning that in order to fully understand these spectra by computational modelling, both electronic and nuclear quantum effects must be included. We show that after reparametrization of the popular CLAYFF model for clays, the O--H stretching region of their vibrational spectra better matches the experimental one, with no detriment to the model's agreement with other experimental properties.
Vibrational spectroscopy is key for probing the interplay between the structure and dynamics of aqueous systems. In order to map different regions of experimental spectra to the microscopic structure of a system, it is important to combine them with first-principles atomistic simulations that incorporate the quantum nature of nuclei. Here, we show that the large cost of calculating quantum vibrational spectra of aqueous systems can be dramatically reduced compared to standard path integral methods by using approximate quantum dynamics based on high-order path integrals. Together with state-of-the-art machine-learned electronic properties, our approach gives an excellent description of the infrared and Raman spectra of bulk water, but also of 2D correlation and more challenging sum-frequency generation spectra of the water-air interface. This paves the way for understanding complex interfaces such as water encapsulated between or in contact with hydrophobic and hydrophilic materials, through robust and inexpensive surface sensitive and multidimensional spectra with first-principles accuracy.
Central to the theory of special cube complexes is Haglund and Wise's construction of the canonical completion and retraction, which enables one to build finite covers of special cube complexes in a highly controlled manner. In this paper we give a new interpretation of this construction using what we call imitator homomorphisms. This provides fresh insight into the construction and enables us to prove various new results about finite covers of special cube complexes -- most of which generalise existing theorems of Haglund--Wise to the non-hyperbolic setting. In particular, we prove a convex version of omnipotence for virtually special cubulated groups.
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.
We study the quasi-isometric rigidity of a large family of finitely generated groups that split as graphs of groups with virtually free vertex groups and two-ended edge groups. Let $G$ be a group that is one-ended, hyperbolic relative to virtually abelian subgroups, and has JSJ decomposition over two-ended subgroups containing only virtually free vertex groups that aren't quadratically hanging. Our main result is that any group quasi-isometric to $G$ is abstractly commensurable to $G$. In particular, our result applies to certain "generic" HNN extensions of a free group over cyclic subgroups.
Leighton's graph covering theorem says that two finite graphs with a common cover have a common finite cover. We present a new proof of this using groupoids, and use this as a model to prove two generalisations of the theorem. The first generalisation, which we refer to as the symmetry-restricted version, restricts how balls of a given size in the universal cover can map down to the two finite graphs when factoring through the common finite cover - this answers a question of Neumann. Secondly, we consider covers of graphs of spaces (or of more general objects), which leads to an even more general version of Leighton's Theorem. We also compute upper bounds for the sizes of the finite covers obtained in Leighton's Theorem and its generalisations. An appendix by Gardam and Woodhouse provides an alternative proof of the symmetry-restricted version, that uses Haar measure instead of groupoids.
Agol proved that hyperbolic cubulated groups are virtually special. The aim of these notes is to make the proof accessible to a wider audience; we retain the underlying ideas and constructions of Agol, but substantially change or add to many parts of the argument to give a more transparent and detailed account.