arXiv subjects
Publications and source records attributed to Yago Antolín.
We characterize, in terms of the defining graph, when a twisted right-angled Artin group (a group whose only relations among pairs of generators are either commuting or Klein-bottle type relations) is left-orderable.
We prove a Tits alternative theorem for subgroups of finitely generated even Artin groups of FC type (EAFC groups), stating that there exists a finite index subgroup such that every subgroup of it is either finitely generated abelian, or maps onto a non-abelian free group. Parabolic subgroups play a key role, and we show that parabolic subgroups of EAFC groups are closed under taking roots.
For Artin groups of dihedral type, we compute the Bredon homology groups of the classifying space for the family of virtually cyclic subgroups with coefficients in the K-theory of a group ring.
We show that the intersection of parabolic subgroups of an even finitely generated FC-type Artin group is again a parabolic subgroup.
We give explicit formulas for the geodesic growth series of a Right Angled Coxeter Group (RACG) based on a link-regular graph that is 4-clique free, i.e. without tetrahedrons
A regular left-order on finitely generated group $G$ is a total, left-multiplication invariant order on $G$ whose corresponding positive cone is the image of a regular language over the generating set of the group under the evaluation map. We show that admitting regular left-orders is stable under extensions and wreath products and give a classification of the groups all whose left-orders are regular left-orders. In addition, we prove that solvable Baumslag-Solitar groups $B(1,n)$ admits a regular left-order if and only if $n\geq -1$. Finally, Hermiller and Sunic showed that no free product admits a regular left-order, however we show that if $A$ and $B$ are groups with regular left-orders, then $(A*B)\times \mathbb{Z}$ admits a regular left-order.
Let $A\neq A_1, A_2, I_{2m}$ be an irreducible Artin--Tits group of spherical type. We show that periodic elements of $A$ and the elements preserving some parabolic subgroup of $A$ act elliptically on the additional length graph $\mathcal{C}_{AL}(A)$, an hyperbolic, infinite diameter graph associated to $A$ constructed by Calvez and Wiest to show that $A/Z(A)$ is acylindrically hyperbolic. We use these results to find an element $g\in A$ such that $\langle P,g \rangle\cong P* \langle g \rangle$ for every proper standard parabolic subgroup $P$ of $A$. The length of $g$ is uniformly bounded with respect to the Garside generators, independently of $A$. This allows us to show that, in contrast with the Artin generators case, the sequence $\{ω(A_n,\mathcal{S})\}_{n\in \mathbb{N}}$ of exponential growth rates of braid groups with respect to the Garside generating set, goes to infinity.
Following ideas that go back to Cannon, we show the rationality of various generating functions of growth sequences counting embeddings of convex subgraphs in locally-finite, vertex-transitive graphs with the (relative) falsification by fellow traveler property (fftp). In particular, we recover results of Cannon, of Epstein, Iano-Fletcher and Zwick, and of Calegari and Fujiwara. One of our applications concerns Schreier coset graphs of hyperbolic groups relative to quasi-convex subgroups, we show that these graphs have rational growth, the falsification by fellow traveler property, and the existence of a lower bound for the growth rate independent of the finite generating set and the infinite index quasi-convex subgroup.
Let $G$ be a Garside group endowed with the generating set $\mathcal{S}$ of non-trivial simple elements, and let $H$ be a parabolic subgroup of $G$. We determine a transversal $T$ of $H$ in $G$ such that each $θ\in T$ is of minimal length in its right-coset, $H θ$, for the word length with respect to $\mathcal{S}$. We show that there exists a regular language $L$ on $\mathcal{S} \cup \mathcal{S}^{-1}$ and a bijection $\mathrm{ev} : L \to T$ satisfying $\mathrm{lg} (U) = \mathrm{lg}_\mathcal{S}( \mathrm{ev}(U))$ for all $U \in L$. From this we deduce that the coset growth series of $H$ in $G$ is rational. Finally, we show that $G$ has fellow projections on $H$ but does not have bounded projections on $H$.
We introduce the space of relative orders on a group and show that it is compact whenever the group is finitely generated. We use this to show that if $G$ is a finitely generated group acting by order preserving homeomorphism of on the line, then if some stabilizer of a point is proper and co-amenable subgroup, then $G$ surjects onto $\mathbb{Z}$. This is a generalization of a theorem of Morris.
We show that a finite collection of stable subgroups of a finitely generated group has finite height, finite width and bounded packing. We then use knowledge about intersections of conjugates to characterize finite families of quasimorphisms on hyperbolically embedded subgroups that can be to simultaneously extended to the ambient group.
We prove that, in a finitely generated residually finite group of subexponential growth, the proportion of commuting pairs is positive if and only if the group is virtually abelian. In particular, this covers the case where the group has polynomial growth (i.e., virtually nilpotent groups, where the hypothesis of residual finiteness is always satisfied). We also show that, for non-elementary hyperbolic groups, the proportion of commuting pairs is always zero.
Following the approach of Dahmani, Guirardel and Osin, we extend the group theoretical Dehn filling theorem to show that the pre-images of infinite order elements have a certain structure of a free product. We then apply this result to show that groups hyperbolic relative to residually finite groups satisfying the Farrell-Jones conjecture, they satisfy the Farrell-Jones conjecture.
Rivin conjectured that the conjugacy growth series of a hyperbolic group is rational if and only if the group is virtually cyclic. Ciobanu, Hermiller, Holt and Rees proved that the conjugacy growth series of a virtually cyclic group is rational. Here we present the proof confirming the other direction of the conjecture, by showing that the conjugacy growth series of a non-elementary hyperbolic group is transcendental. We also present and prove some variations of Rivin's conjecture for commensurability classes and primitive conjugacy classes. We then explore Rivin's conjecture for finitely generated acylindrically hyperbolic groups and prove a formal language version of it, namely that no set of minimal length conjugacy representatives can be unambiguous context-free.
Modelled on efficient algorithms for solving the conjugacy problem in hyperbolic groups, we define and study the permutation conjugacy length function. This function estimates the length of a short conjugator between words $u$ and $v$, up to taking cyclic permutations. This function might be bounded by a constant, even in the case when the standard conjugacy length function is unbounded. We give applications to the complexity of the conjugacy problem, estimating conjugacy growth rates, and languages. Our main result states that for a relatively hyperbolic group, the permutation conjugacy length function is bounded by the permutation conjugacy length function of the parabolic subgroups.
Given a finitely generated relatively hyperbolic group $G$, we construct a finite generating set $X$ of $G$ such that $(G,X)$ has the `falsification by fellow traveler property' provided that the parabolic subgroups $\{H_ω\}_{ω\in Ω}$ have this property with respect to the generating sets $\{X\cap H_ω\}_{ω\in Ω}$. This implies that groups hyperbolic relative to virtually abelian subgroups, which include all limit groups and groups acting freely on $\mathbb{R}^n$-trees, or geometrically finite hyperbolic groups, have generating sets for which the language of geodesics is regular, and the complete growth series and complete geodesic series are rational. As an application of our techniques, we prove that if each $H_ω$ admits a geodesic biautomatic structure over $X\cap H_ω$, then $G$ has a geodesic biautomatic structure. Similarly, we construct a finite generating set $X$ of $G$ such that $(G,X)$ has the `bounded conjugacy diagrams' property or the `neighbouring shorter conjugate' property if the parabolic subgroups $\{H_ω\}_{ω\in Ω}$ have this property with respect to the generating sets $\{X\cap H_ω\}_{ω\in Ω}$. This implies that a group hyperbolic relative to abelian subgroups has a generating set for which its Cayley graph has bounded conjugacy diagrams, a fact we use to give a cubic time algorithm to solve the conjugacy problem. Another corollary of our results is that groups hyperbolic relative to virtually abelian subgroups have a regular language of conjugacy geodesics.
Let G be a group and H be a subgroup of G. We say that H is left relatively convex in G if the left G-set G/H has at least one G-invariant order; when G is left orderable, this holds if and only if H is convex in G under some left ordering of G. We give a criterion for H to be left relatively convex in G that generalizes a famous theorem of Burns and Hale and has essentially the same proof. We show that all maximal cyclic subgroups are left relatively convex in free groups, in right-angled Artin groups, and in surface groups that are not the Klein-bottle group. The free-group case extends a result of Duncan and Howie. We show that if G is left orderable, then each free factor of G is left relatively convex in G. More generally, for any graph of groups, if each edge group is left relatively convex in each of its vertex groups, then each vertex group is left relatively convex in the fundamental group; this generalizes a result of Chiswell. We show that all maximal cyclic subgroups in locally residually torsion-free nilpotent groups are left relatively convex.
Let $n\in \mathbb{N}$. Houghton's group $H_n$ is the group of permutations of $\{1,\dots, n\}\times \mathbb{N}$, that eventually act as a translation in each copy of $\mathbb{N}$. We prove the solvability of the conjugacy problem and conjugator search problem for $H_n$, $n\geq 2$.