arXiv subjects

Francis Wagner

Publications and source records attributed to Francis Wagner.

7 recordsLinked to original sources

arXiv2026

Conjugator Lengths and Isoperimetric functions

We show that for any recognizing $S$-machine $\textbf{S}$ with superadditive time function $f$, there exists a finitely presented group whose conjugator length function is quadratic and whose Dehn function grows like the square of $f$. This result is dual to that of a previous paper of the authors, which constructed a family of groups with cubic Dehn function that have various conjugator length functions. We thereby give the first known example of a finitely presented group whose conjugator length function is recursive, but whose Word and Conjugacy Problems are both undecidable. This answers an analogue of a question of Rips. Moreover, given a finitely presented group $G$ with decidable Word Problem, we obtain a finitely presented group with decidable Conjugacy Problem whose Dehn function grows faster than that of $G$. Finally, combining this result with its dual, for a wide array of pairs of functions $(f,g)$ we furnish an example of a finitely presented group with Dehn function equivalent to $f$ and conjugator length function equivalent to $g$. This shows that the two invariants are very strongly independent and making significant progress on a question of Bridson, Riley, and Sale.

math.GR↗

arXiv2026

Conjugator Length in Finitely Presented Groups

The conjugator length function of a finitely generated group is the function $f$ so that $f(n)$ is the minimal upper bound on the length of a word realizing the conjugacy of two words of length at most $n$. We study herein the spectrum of functions which can be realized as the conjugator length function of a finitely presented group, showing that it contains every function that can be realized as the Dehn function of a finitely presented group. In particular, given a real number $\alpha\geq2$ which is computable in double-exponential time, we show there exists a finitely presented group whose conjugator length function is asymptotically equivalent to $n^\alpha$. This yields a substantial refinement to results of Bridson and Riley. We attain this result through the computational model of $S$-machines, achieving the more general result that any sufficiently large function which can be realized as the time function of an $S$-machine can also be realized as the conjugator length function of a finitely presented group. Finally, we use the constructed groups to explore the relationship between the conjugator length function, the Dehn function, and the annular Dehn function in finitely presented groups.

math.GR↗

arXiv2025

Quasi-isometric Higman embeddings and the Dehn function

This is the first of a sequence of papers devoted to studying the link between the complexity of the Word Problem for a finitely generated recursively presented group $G$ and the isoperimetric functions of the finitely presented groups in which $G$ embeds. We prove here that if a finitely generated group has a presentation $\mathcal{P}$ whose relators can be enumerated by a computational model satisfying certain technical requirements, then the group embeds quasi-isometrically into a finitely presented group whose Dehn function is bounded above by a function of the model's computational complexity and the Dehn function of $\mathcal{P}$. This generalizes a previous result of the author pertaining to the embeddings of free Burnside groups and gives a recipe for establishing such Higman embeddings into groups with desired geometric properties. As an example of the use of this embedding scheme, we find a substantial improvement to the seminal result of Birget, Ol'shanskii, Rips, and Sapir showing that the Word Problem of a finitely generated group is in class NP if and only if the group embeds into a finitely presented group with polynomial Dehn function.

math.GR↗

arXiv2024

Malnormal Subgroups of Finitely Presented Groups

The following refinement of the Higman embedding theorem is proved: A finitely generated group $R$ is recursively presented if and only if there exists a quasi-isometric malnormal embedding of $R$ into a finitely presented group $H$ such that the image of the embedding enjoys the congruence extension property. Moreover, it is shown that the finitely presented group $H$ can be constructed to have decidable Word Problem if and only if the Word Problem for $R$ is decidable, yielding a refinement of a theorem of Clapham. Finally, given a countable group $G$ and a computable function $\ell:G\to\mathbb{N}$ satisfying some necessary requirements, it is proved that there exists a malnormal embedding of $G$ into a finitely presented group $H$ such that the restriction of $|\cdot|_H$ to $G$ is equivalent to $\ell$, producing a refinement of a theorem of Ol'shanskii.

math.GR↗

arXiv2023

Quasilinear Emulation of Turing Machines by S-machines

We prove that for any $\varepsilon>0$, a non-deterministic Turing machine $\mathcal{T}$ with time complexity $T(n)$ can be emulated by an $S$-machine with time and space complexities at most $T(n)^{1+\varepsilon}$ and $T(n)$, respectively. This improves the bounds on the emulation in arXiv:math/9811105 and leads to improved bounds in the main theorem of arXiv:math/9811106. In particular, for a non-hyperbolic finitely generated group $G$ whose word problem has linear time complexity, this yields an embedding of $G$ into a finitely presented group $H$ such that $G$ has bounded distortion in $H$ and the Dehn function of $G$ in $H$ is bounded above by $n^{2+\varepsilon}$, an optimal bound modulo the $\varepsilon$ factor. As a means to this end, we introduce and develop the theory of $S$-graphs, giving a different perspective on the construction of $S$-machines akin to a crude object-oriented programming language.

math.GR↗

arXiv2020

Torsion Subgroups of Groups with Quadratic Dehn Function

We construct the first examples of finitely presented groups with quadratic Dehn function containing a finitely generated infinite torsion subgroup. These examples are "optimal" in the sense that the Dehn function of any such finitely presented group must be at least quadratic. Moreover, we show that for any $n\geq2^{48}$ such that $n$ is either odd or divisible by $2^9$, any infinite free Burnside group with exponent $n$ is a quasi-isometrically embedded subgroup of a finitely presented group with quadratic Dehn function satisfying the Congruence Extension Property.

math.GR↗

arXiv2020

Torsion Subgroups of Groups with Cubic Dehn Function

We construct the first examples of finitely presented groups with cubic Dehn function containing a finitely generated infinite torsion subgroup. Moreover, we show that any infinite free Burnside group with sufficiently large odd exponent can be embedded as a subgroup of a finitely presented group with cubic Dehn function.

math.GR↗