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Robert Alonzo Lyman

Publications and source records attributed to Robert Alonzo Lyman.

10 recordsLinked to original sources
arXiv2026

Coarse geometry of homeomorphism groups: Classifying countable Stone spaces

Towards developing the tools of geometric group theory for non-locally compact topological groups, we give one of the first complete classifications of a family of such groups up to coarse equivalence, and when possible, up to quasi-isometry. In a previous paper, we placed the homeomorphism groups of countable Stone spaces into three classes: coarsely bounded, unbounded yet generated by a coarsely bounded set, and unbounded but not generated by any coarsely bounded set. Now we show that these are the coarse equivalence classes: Any two groups within one of these classes are in fact coarsely equivalent. Furthermore, we show that groups in the second class are quasi-isometric to the Hamming cube, the space comprising infinite binary sequences with finitely many nonzero entries equipped with the Hamming distance, and that groups in the third class are coarsely equivalent to the set of leaves of the regular one-ended tree of countably infinite valence. As part of the proof, we show that infinite Hamming graphs over finite alphabets are all bi-Lipschitz equivalent and prove a coarse geometric classification result for topological groups admitting exhaustions by proper, open, coarsely bounded subgroups.

math.GR↗
arXiv2026

Finiteness properties of stabilisers of oligomorphic actions

An action of a group on a set is oligomorphic if it has finitely many orbits of $n$-element subsets for all $n$. We prove that for a large class of groups (including all groups of finite virtual cohomological dimension and all countable linear groups), for any oligomorphic action of such a group on an infinite set there exists a finite subset whose stabiliser is not of type $\mathrm{FP}_\infty$. This leads to obstructions on finiteness properties for permutational wreath products and twisted Brin-Thompson groups. We also prove a version for actions on flag complexes, and discuss connections to the Boone-Higman conjecture. In the appendix, we improve on the criterion of Bartholdi-Cornulier-Kochloukova for finiteness properties of wreath products, and the criterion of Kropholler-Martino for finiteness properties of graph-wreath products.

math.GR↗
arXiv2025

On a lemma of Milnor and Schwarz, après Rosendal

Perhaps the fundamental theorem of geometric group theory, the Milnor--Schwarz lemma gives conditions under which the orbit map relating the geometry of a geodesic metric space and the word metric on a group acting isometrically on the space is a quasi-isometry. Pioneering work of Rosendal makes these and other techniques of geometric group theory applicable to an arbitrary (topological) group. We give a succinct treatment of the Milnor--Schwarz lemma, setting it within this context. We derive some applications of this theory to non-Archimedean groups, which have plentiful continuous actions on graphs. In particular, we sharpen results of BarNatan and Verberne on actions of "big" mapping class groups on hyperbolic graphs and clarify a project begun by Mann and Rafi to classify these mapping class groups up to quasi-isometry, noting some extensions to the theory of mapping class groups of locally finite infinite graphs and homeomorphism groups of Stone spaces.

math.GR↗
arXiv2025

Graphical models for topological groups: A case study on countable Stone spaces

By analogy with the Cayley graph of a group with respect to a finite generating set or the Cayley--Abels graph of a totally disconnected, locally compact group, we detail countable connected graphs associated to Polish groups that we term Cayley--Abels--Rosendal graphs. A group admitting a Cayley--Abels--Rosendal graph acts on it continuously, coarsely metrically properly and cocompactly by isometries of the path metric. By an expansion of the Milnor--Schwarz lemma, it follows that the group is generated by a coarsely bounded set and the group equipped with a word metric with respect to a coarsely bounded generating set and the graph are quasi-isometric. In other words, groups admitting Cayley--Abels--Rosendal graphs are topological analogues of finitely generated groups. Our goal is to introduce this topological perspective on the work of Rosendal to a geometric group theorist. We apply these concepts to homeomorphism groups of countable Stone spaces. We completely characterize when these homeomorphism groups are coarsely bounded, when they are locally bounded (all of them are), and when they admit a Cayley--Abels--Rosendal graph, and if so produce a coarsely bounded generating set.

math.GR↗
arXiv2025

Conformal dimension bounds for certain Coxeter group Bowditch boundaries

We give upper and lower bounds on the conformal dimension of the Bowditch boundary of a Coxeter group with defining graph a complete graph and edge labels at least three. The lower bounds are obtained by quasi-isometrically embedding Gromov's round trees in the Davis complex. The upper bounds are given by exhibiting a geometrically finite action on a CAT(-1) space and bounding the Hausdorff dimension of the visual boundary of this space. Our results imply that there are infinitely many quasi-isometry classes within each infinite family of such Coxeter groups with edge labels bounded from above. As an application, we prove there are infinitely many quasi-isometry classes among the family of hyperbolic groups with Pontryagin sphere boundary. Combining our results with work of Bourdon--Kleiner proves the conformal dimension of the boundaries of hyperbolic groups in this family achieves a dense set in $(1,\infty)$.

math.GT↗
arXiv2025

The complex of cuts in a Stone space

Stone's representation theorem asserts a duality between Boolean algebras on the one hand and Stone space, which are compact, Hausdorff, and totally disconnected, on the other. This duality implies a natural isomorphism between the homeomorphism group of the space and the automorphism group of the algebra. We introduce a complex of cuts on which these groups act, and prove that when the algebra is countable and the space has at least five points, that these groups are the full automorphism group of the complex.

math.GT↗
arXiv2025

When is the Outer Space of a free product CAT(0)?

Generalizing Culler and Vogtmann's Outer Space for the free group, Guirardel and Levitt construct an Outer Space for a free product of groups. We completely characterize when this space (or really its simplicial spine) supports an equivariant piecewise-Euclidean or piecewise-hyperbolic CAT(0) metric. Our results are mostly negative, extending thesis work of Bridson and related to thesis work of Cunningham. In particular, provided the dimension of the spine is at least three, it is never CAT(0). Surprisingly, we exhibit one family of free products for which the Outer Space is two-dimensional and does support an equivariant CAT(0) metric.

math.GR↗