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George Domat

Publications and source records attributed to George Domat.

14 recordsLinked to original sources
arXiv2026

Asymptotically conformal and asymptotically rigid mapping class groups

We give conditions ensuring that an asymptotically rigid mapping class group, specifically a surface Houghton group $\mathcal{H}(S)$, has finite index in the asymptotically conformal modular group $\text{Mod}_0(X)$, where $X$ is a hyperbolic structure on $S$. These include geometric conditions on the pieces of the underlying rigid structure, as well as the existence in $\mathcal{H}(S)$ of an end-periodic homeomorphism which is asymptotically conformal. As a consequence, if $S$ has $n\geq 3$ ends, then $\text{Mod}_0(X)$ has type $F_{n-1}$ but not $FP_n$. We also establish analogous results for $L^p$ modular groups.

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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.

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arXiv2025

The infinite dimensional geometry of conjugation invariant generating sets

We consider a number of examples of groups together with an infinite conjugation invariant generating set, including: the free group with the generating set of all separable elements; surface groups with the generating set of all non-filling curves; mapping class groups and outer automorphism groups of free groups with the generating sets of all reducible elements; and groups with suitable actions on Gromov hyperbolic spaces with a generating set of elliptic elements. Building on work of Brandenbursky-Gal-K\c{e}dra-Marcinkowski, in these Cayley graphs we show that there are quasi-isometrically embedded copies of $\mathbb{Z}^m$ for all $m\geq 1$. A corollary is that these Cayley graphs have infinite asymptotic dimension. By additionally building a new subsurface projection analogue for the free splitting graph, which is valued in the above Cayley graph of the free group and may be of independent interest, we are able to recover Sabalka-Savchuk's result that the edge-splitting graph of the free group has quasi-isometrically embedded copies of $\mathbb{Z}^m$ for all $m\geq 1$.

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arXiv2024

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.

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arXiv2024

Classification of Stable Surfaces with respect to Automatic Continuity

We provide a complete classification of when the homeomorphism group of a stable surface, $\Sigma$, has the automatic continuity property: Any homomorphism from Homeo$(\Sigma)$ to a separable group is necessarily continuous. This result descends to a classification of when the mapping class group of $\Sigma$ has the automatic continuity property. Towards this classification, we provide a general framework for proving automatic continuity for groups of homeomorphisms. Applying this framework, we also show that the homeomorphism group of any stable second countable Stone space has the automatic continuity property. Under the presence of stability this answers two questions of Mann.

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arXiv2023

Generating Sets and Algebraic Properties of Pure Mapping Class Groups of Infinite Graphs

We completely classify the locally finite, infinite graphs with pure mapping class groups admitting a coarsely bounded generating set. We also study algebraic properties of the pure mapping class group: We establish a semidirect product decomposition, compute first integral cohomology, and classify when they satisfy residual finiteness and the Tits alternative. These results provide a framework and some initial steps towards quasi-isometric and algebraic rigidity of these groups.

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arXiv2023

Multitwists in big mapping class groups

We show that the closure of the compactly supported mapping class group of an infinite-type surface is not generated by the collection of multitwists (i.e. products of powers of twists about disjoint non-accumulating curves).

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arXiv2022

Thurston's Theorem: Entropy in Dimension One

In his paper, Thurston shows that a positive real number $h$ is the topological entropy for an ergodic traintrack representative of an outer automorphism of a free group if and only if its expansion constant $\lambda = e^h$ is a weak Perron number. This is a powerful result, answering a question analogous to one regarding surfaces and stretch factors of pseudo-Anosov homeomorphisms. However, much of the machinery used to prove this seminal theorem on traintrack maps is contained in the part of Thurston's paper on the entropy of postcritically finite interval maps and the proof difficult to parse. In this expository paper, we modernize Thurston's approach, fill in gaps in the original paper, and distill Thurston's methods to give a cohesive proof of the traintrack theorem. Of particular note is the addition of a proof of ergodicity of the traintrack representatives, which was missing in Thurston's paper.

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arXiv2022

Coarse Geometry of Pure Mapping Class Groups of Infinite Graphs

We discuss the large-scale geometry of pure mapping class groups of locally finite, infinite graphs, motivated by recent work of Algom-Kfir--Bestvina and the work of Mann--Rafi on the large-scale geometry of mapping class groups of infinite-type surfaces. Using the framework of Rosendal for coarse geometry of non-locally compact groups, we classify when the pure mapping class group of a locally finite, infinite graph is globally coarsely bounded (an analog of compact) and when it is locally coarsely bounded (an analog of locally compact). Our techniques give lower bounds on the first integral cohomology of the pure mapping class group for some graphs and allow us to compute the asymptotic dimension of all locally coarsely bounded pure mapping class groups of infinite rank graphs. This dimension is always either zero or infinite.

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arXiv2022

Free products from spinning and rotating families

The far-reaching work of Dahmani-Guirardel-Osin and recent work of Clay-Mangahas-Margalit provide geometric approaches to the study of the normal closure of a subgroup (or a collection of subgroups)in an ambient group $G$. Their work gives conditions under which the normal closure in $G$ is a free product. In this paper we unify their results and simplify and significantly shorten the proof of the Dahmani-Guirardel-Osin theorem.

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arXiv2021

Big pure mapping class groups are never perfect

We show that the closure of the compactly supported mapping class group of an infinite type surface is not perfect and that its abelianization contains a direct summand isomorphic to an uncountable direct sum of rationals. We also extend this to the Torelli group and show that in the case of surfaces with infinite genus the abelianization of the Torelli group contains an indivisible copy of an uncountable free abelian group as well. Finally we give an application to the question of automatic continuity by exhibiting discontinuous homomorphisms to the rationals.

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