A mathematical preprint dated 26 August 2026 proves that a Vietoris-Rips complex at parameter 2 is contractible for a restricted family of right-angled Artin groups, provided the defining graph is triangle-free and the standard word metric is used. It also develops a local test that gives sufficient conditions for connectivity and contractibility in a broader setting.
The result concerns abstract mathematical structures. The paper analyzes finitely generated groups, their word metrics and finite subsets used to form Vietoris-Rips complexes, rather than an empirical dataset.
A global question approached through local structure
The work begins with a broad question: whether a finitely generated group with a proper cocompact action on a contractible simplicial complex must have some finite-parameter Vietoris-Rips complex that is contractible. The paper does not settle that question for every group. Instead, it develops conditions that can be checked around finite subsets and applies them to a specific family of groups.
The proof works with finite subsets selected by a diameter bound at a chosen parameter. It then uses distance to a fixed point as an initial measurement, with word length serving as the corresponding measurement for finitely generated groups.
The local test
The approach uses discrete Morse theory, which assigns a structured score to finite subsets and examines the resulting levels. The initial measurement is distance to a fixed point; for a finitely generated group, the corresponding measurement is word length.
On admissible finite subsets, the paper defines the function f(S)=len_min(S)+|S|/N, where N is a uniform strict upper bound, and proves that this function is a Morse function. The formula combines the shortest relevant length in the set with a term that records the set's size.
Descending links decompose as joins of descending face links and descending coface links. When length is not constant on S, the face link, and therefore the full descending link, is contractible. The remaining cases are those in which length is constant.
The central criterion is sufficient rather than universal. For every nonempty finite set S other than the identity singleton, when the set has diameter at most t and constant length, the paper examines a descending coface complex. If these local complexes are (n−|S|−1)-connected, then VR_t(G) is (n−1)-connected. If all of them are contractible, then VR_t(G) is contractible.
What happens in triangle-free graph groups
The main application concerns right-angled Artin groups, or RAAGs, defined by graphs. The paper calls a RAAG two-dimensional when its defining graph has no 3-cliques, and also describes that case as triangle-free.
Under that graph condition, every finite set S with at least 2 elements, pairwise distance 2 and constant word length has a unique common descending neighbor.
That result supplies the needed coface calculation for non-singleton sets: when the length is constant and the common descending neighbor is unique, the descending coface link is contractible. For singleton sets S={g} with g different from the identity, the descending coface link is also contractible.
Taken together with the local criterion, these results give the paper's main application: VR_2(A_Γ) is contractible whenever Γ is triangle-free, with the standard word metric. The statement is confined to that graph condition, metric and parameter.
A separate route to finite presentability
The paper also gives sufficient conditions for finite presentability, meaning a finite description of a group by generators and relations. One route says that, for some t, every nonidentity element g must have a connected descending complex, while every equal-length pair at distance t must have a nonempty descending complex.
A second route applies when the Cayley graph has no odd-length cycles. In that case, connected singleton descending complexes for every nonidentity g at some odd t are sufficient for finite presentability.
The finite-presentability conclusion comes from proving that VR_t(G) is simply connected and then using the geometric action of G on that complex. These are sufficient conditions, not a necessary-and-sufficient characterization of finite presentability.
A result with a clear boundary
The preprint's conclusions depend on their stated hypotheses, including finite generation, the relevant word metric, diameter restrictions and the required descending-link properties. Its contractibility result for RAAGs is specifically limited to triangle-free defining graphs and the standard word metric.
The paper does not establish that every group in its opening question has a contractible Vietoris-Rips complex at a finite parameter. It also does not establish the result for all right-angled Artin groups, for every parameter or outside the stated metric and graph setting.
The evidence consists of mathematical definitions, lemmas, propositions and theorem proofs, so there are no empirical outcome estimates or statistical uncertainty to report. The document is labeled arXiv version 1 and dated 26 August 2026.
The acknowledgments mention work appearing in the first author's PhD thesis, but no funding source is reported.
Paper data and sources
Original title: Word length, Morse theory, and Vietoris-Rips complexes
Authors: Seth Hulbert, Matthew C. B. Zaremsky
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
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