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Iván Sadofschi Costa

Publications and source records attributed to Iván Sadofschi Costa.

9 recordsLinked to original sources
arXiv2025

Group actions on contractible $2$-complexes I

In this series of two articles, we prove that every action of a finite group $G$ on a finite and contractible $2$-complex has a fixed point. The proof goes by constructing a nontrivial representation of the fundamental group of each of the acyclic $2$-dimensional $G$-complexes constructed by Oliver and Segev. In the first part we develop the necessary theory and cover the cases where $G=\mathrm{PSL}_2(2^n)$, $G=\mathrm{PSL}_2(q)$ with $q\equiv 3\pmod 8$ or $G=\mathrm{Sz}(2^n)$. The cases $G=\mathrm{PSL}_2(q)$ with $q\equiv 5\pmod 8$ are addressed in the second part.

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arXiv2025

Group actions on contractible $2$-complexes II

In this second part we prove that, if $G$ is one of the groups $\mathrm{PSL}_2(q)$ with $q>5$ and $q\equiv 5\pmod {24}$ or $q\equiv 13 \pmod{24}$, then the fundamental group of every acyclic $2$-dimensional, fixed point free and finite $G$-complex admits a nontrivial representation in a unitary group $\mathrm{U}(m)$. This completes the proof of the following result: every action of a finite group on a finite and contractible $2$-complex has a fixed point.

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arXiv2020

Acyclic $2$-dimensional complexes and Quillen's conjecture

Let $G$ be a finite group and $\mathcal{A}_p(G)$ be the poset of nontrivial elementary abelian $p$-subgroups of $G$. Quillen conjectured that $O_p(G)$ is nontrivial if $\mathcal{A}_p(G)$ is contractible. We prove that $O_p(G)\neq 1$ for any group $G$ admitting a $G$-invariant acyclic $p$-subgroup complex of dimension $2$. In particular, it follows that Quillen's conjecture holds for groups of $p$-rank $3$. We also apply this result to establish Quillen's conjecture for some particular groups not considered in the seminal work of Aschbacher--Smith.

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arXiv2020

Generalized small cancellation conditions, non-positive curvature and diagrammatic reducibility

We present a metric condition ${\LARGEτ}'$ which describes the geometry of classical small cancellation groups and applies also to other known classes of groups such as two-dimensional Artin groups. We prove that presentations satisfying condition ${\LARGEτ}'$ are diagrammatically reducible in the sense of Sieradski and Gersten. In particular we deduce that the standard presentation of an Artin group is aspherical if and only if it is diagrammatically reducible. We show that, under some extra hypotheses, ${\LARGEτ}'$-groups have quadratic Dehn functions and solvable conjugacy problem. In the spirit of Greendlinger's lemma, we prove that if a presentation $P=\langle X \mid R\rangle$ of group $G$ satisfies conditions ${\LARGEτ}'-C'(\frac{1}{2})$, the length of any nontrivial word in the free group generated by $X$ representing the trivial element in $G$ is at least that of the shortest relator. We also introduce a strict metric condition ${\LARGEτ}'_{<}$, which implies hyperbolicity. Finally we investigate non-metric and dual variants of these conditions, and study a harmonic mean version of small cancellation theory.

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arXiv2017

The complex of partial bases of a free group

We prove that the simplicial complex whose simplices are the nonempty partial bases of $\mathbb{F}_n$ is homotopy equivalent to a wedge of $(n-1)$-spheres. Moreover, we show that it is Cohen-Macaulay.

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arXiv2016

On a generalization of the seating couples problem

We prove a conjecture of Adamaszek generalizing the seating couples problem to the case of $2n$ seats. Concretely, we prove that given a positive integer $n$ and $d_1,\ldots,d_n\in(\mathbb{Z}/2n)^*$ we can partition $\mathbb{Z}/2n$ into $n$ pairs with differences $d_1,\ldots,d_n$.

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arXiv2015

Presentation complexes with the fixed point property

We prove that there exists a compact two-dimensional polyhedron with the fixed point property and even Euler characteristic. This answers a question posed by R.H. Bing in 1969. We also settle another of Bing's questions.

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