Tim Riley
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View article: Linear Diophantine equations and conjugator length in 2-step nilpotent groups
Linear Diophantine equations and conjugator length in 2-step nilpotent groups Open
We establish upper bounds on the lengths of minimal conjugators in 2-step nilpotent groups. These bounds exploit the existence of small integral solutions to systems of linear Diophantine equations. We prove that in some cases these bounds…
View article: Conjugacy in a family of free-by-cyclic groups
Conjugacy in a family of free-by-cyclic groups Open
We analyse the geometry and complexity of the conjugacy problem in a family of free-by-cyclic groups $H_m=F_m\rtimes\mathbb{Z}$ where the defining free-group automorphism is positive and polynomially growing. We prove that the conjugator l…
View article: The lengths of conjugators in the model filiform groups
The lengths of conjugators in the model filiform groups Open
The conjugator length function of a finitely generated group $Γ$ gives the optimal upper bound on the length of a shortest conjugator for any pair of conjugate elements in the ball of radius $n$ in the Cayley graph of $Γ$. We prove that po…
View article: Exponentially Distorted Subgroups in Wreath Products
Exponentially Distorted Subgroups in Wreath Products Open
We exhibit exponentially distorted subgroups in $\mathbb{Z} \wr ( \mathbb{Z} \wr \mathbb{Z} )$ and $\mathbb{Z} \wr F_2$.
View article: Dehn functions of mapping tori of right-angled Artin groups
Dehn functions of mapping tori of right-angled Artin groups Open
The algebraic mapping torus $M_{\Phi }$ of a group $G$ with an automorphism $\Phi$ is the HNN-extension of $G$ in which conjugation by the stable letter performs $\Phi$ . We classify the Dehn functions of $M_{\Phi }$ in terms of $\Phi$ for…
View article: Rapid stellar and binary population synthesis with COMPAS
Rapid stellar and binary population synthesis with COMPAS Open
Compact Object Mergers: Population Astrophysics and Statistics (COMPAS; https://compas.science) is a public rapid binary population synthesis code. COMPAS generates populations of isolated stellar binaries under a set of parametrized assum…
View article: Cannon–Thurston maps, subgroup distortion, and hyperbolic hydra
Cannon–Thurston maps, subgroup distortion, and hyperbolic hydra Open
There is a family of hyperbolic groups known as hyperbolic hydra which contain heavily distorted free subgroups. We prove the existence of Cannon–Thurston maps (that is, maps of the boundaries induced by subgroup inclusion) for these free …
View article: Dehn functions of mapping tori of right-angled Artin groups
Dehn functions of mapping tori of right-angled Artin groups Open
The algebraic mapping torus $M_Φ$ of a group $G$ with an automorphism $Φ$ is the HNN-extension of $G$ in which conjugation by the stable letter performs $Φ$. We classify the Dehn functions of $M_Φ$ in terms of $Φ$ for a number of right-ang…
View article: Soficity and variations on Higman’s group
Soficity and variations on Higman’s group Open
A group is sofic when every finite subset can be well approximated in a finite symmetric group. No example of a non-sofic group is known. Higman's group, which is a circular amalgamation of four copies of the Baumslag–Solitar group, is a c…
View article: Ethnohistoric Records of Hunter-Gatherer Diet of the Texas/Mexico Borderlands: Implications for Staple Foods of the Lower Pecos Canyonlands During the Holocene
Ethnohistoric Records of Hunter-Gatherer Diet of the Texas/Mexico Borderlands: Implications for Staple Foods of the Lower Pecos Canyonlands During the Holocene Open
Hinds Cave (41VV456) and other rockshelters excavated in the Lower Pecos Canyonlands have yielded thousands of coprolites spanning the Holocene. To date, several hundred specimens have been analyzed, providing a detailed record of meals co…
View article: Computing area in presentations of the trivial group
Computing area in presentations of the trivial group Open
We give polynomial-time dynamic-programming algorithms finding the areas of words in the presentations $\langle a, b \mid a, b \rangle$ and $\langle a, b \mid a^k, b^k; \ k \in \mathbb {N}\rangle$ of the trivial group. In the first of thes…
View article: Computing area in presentations of the trivial group
Computing area in presentations of the trivial group Open
We give polynomial-time dynamic-programming algorithms finding the areas of words in the presentations $\langle a, b \mid a, b \rangle$ and $\langle a, b \mid a^k, b^k; \ k \in \mathbb{N} \rangle$ of the trivial group. In the first of thes…
View article: Computing area in group presentations
Computing area in group presentations Open
The width of a word $w$ representing an element in a free group $F(a,b)$ was defined by Jiang to be the minimal $N$ such that $w$ freely equals a product of $N$ conjugates of powers of $a$ and $b$. In 1991 Grigorchuk and Kurchanov gave an …
View article: Ackermannian Integer Compression and the Word Problem for Hydra Groups
Ackermannian Integer Compression and the Word Problem for Hydra Groups Open
For a finitely presented group, the word problem asks for an algorithm which declares whether or not words on the generators represent the identity. The Dehn function is a complexity measure of a direct attack on the word problem by applyi…
View article: Ackermannian Integer Compression and the Word Problem for Hydra Groups.
Ackermannian Integer Compression and the Word Problem for Hydra Groups. Open
For a finitely presented group, the word problem asks for an algorithm which declares whether or not words on the generators represent the identity. The Dehn function is a complexity measure of a direct attack on the word problem by applyi…