Recent zbMATH articles in MSC 57Mhttps://zbmath.org/atom/cc/57M2024-02-15T19:53:11.284213ZWerkzeugHigher rank confining subsets and hyperbolic actions of solvable groupshttps://zbmath.org/1526.200582024-02-15T19:53:11.284213Z"Abbott, Carolyn R."https://zbmath.org/authors/?q=ai:abbott.carolyn-r"Balasubramanya, Sahana H."https://zbmath.org/authors/?q=ai:balasubramanya.sahana-h"Rasmussen, Alexander J."https://zbmath.org/authors/?q=ai:rasmussen.alexander-jIn some recent papers, the authors have described the hyperbolic actions of several families of solvable groups (see e.g. [Algebr. Geom. Topol. 23, No. 4, 1641--1692 (2023; Zbl 1522.20162)] by the first and the third author). A key tool for these investigations is the machinery of confining subsets of \textit{P.-E. Caprace} et al. [J. Eur. Math. Soc. (JEMS) 17, No. 11, 2903--2947 (2015; Zbl 1330.43002)], which applies, in particular, to solvable groups with virtually cyclic abelianizations.
In the paper under review, the authors extend this machinery and give a correspondence between the hyperbolic actions of certain solvable groups with higher-rank abelianizations and confining subsets of these more general groups. They apply this extension to give a complete description of the hyperbolic actions of generalized solvable Baumslag-Solitar groups and to reprove a result of \textit{W. Sgobbi} and \textit{P. Wong} [Commun. Algebra 51, No. 8, 3354--3370 (2023; Zbl 07700055)] computing their Bieri-Neumann-Strebel invariants.
Reviewer: Egle Bettio (Venezia)Quasi-isometry invariance of relative filling functions (with an appendix by Ashot Minasyan)https://zbmath.org/1526.200602024-02-15T19:53:11.284213Z"Hughes, Sam"https://zbmath.org/authors/?q=ai:hughes.sam"Martínez-Pedroza, Eduardo"https://zbmath.org/authors/?q=ai:martinez-pedroza.eduardo"Saldaña, Luis Jorge Sánchez"https://zbmath.org/authors/?q=ai:sanchez-saldana.luis-jorgeLet \((G,\mathcal{P})\) be a pair where \(G\) is a finitely generated group with a chosen word metric \(\mathsf{dist}_{G}\) and \(\mathcal{P}\) is a finite collection of subgroups. Let \(\mathsf{hdist}_{G}\) denote the Hausdorff distance between subsets of \(G\) and let \(G/\mathcal{P}\) denote the collection of left cosets \(gP\) for \(g \in G\) and \(P\in \mathcal{P}\). For constants \(L\geq 1\), \(C \geq 0\) and \(M \geq 0\), an \(( L, C, M)\)-quasi-isometry of pairs \(q: (G, \mathcal{P}) \rightarrow (H,\mathcal{Q})\) is an \((L, C)\)-quasi-isometry \(q: G \rightarrow H\) such that the relation \(\{ (A,B) \in G/\mathcal{P} \times H/\mathcal{Q} \mid \mathsf{hdist}_{H}(q(A), B) < M \}\) satisfies the condition that the projections to \(G/\mathcal{P}\) and \(H/\mathcal{Q}\) are surjective.
In the paper under review, the authors prove that the relative Dehn function of a pair \((G,\mathcal{P})\) is invariant under quasi-isometry of pairs. Along the way, they show quasi-isometries of pairs preserve almost malnormality of the collection and fineness of the associated coned-off Cayley graphs. They also prove that for a cocompact simply connected combinatorial \(G-2\)-complex \(X\) with finite edge stabilisers, the combinatorial Dehn function is well defined if and only if the 1-skeleton of \(X\) is fine. Finally, the authors prove that if \(H\) is a hyperbolically embedded subgroup of a finitely presented group \(G\), then the relative Dehn function of the pair \((G, \{H\})\) is well defined.
Reviewer: Egle Bettio (Venezia)Dehn functions of coabelian subgroups of direct products of groupshttps://zbmath.org/1526.200612024-02-15T19:53:11.284213Z"Kropholler, Robert"https://zbmath.org/authors/?q=ai:kropholler.robert-p"Llosa Isenrich, Claudio"https://zbmath.org/authors/?q=ai:llosa-isenrich.claudioIn geometric group theory, a Dehn function is an optimal function associated to a finite group presentation which bounds the area of a relation in that group in terms of the length of that relation. The growth type of the Dehn function is a quasi-isometry invariant of a finitely presented group. The definition of a Dehn function depends on the choice of finite presentation for the group. However, it is well known that its asymptotic equivalence class does not.
A group \(G\) is of finiteness type \(\mathcal{F}_{k}\) if it admits a \(CW\)-complex \(K(G,1)\) with finitely many cells of dimension \(\leq k\).
A quantitative approach to understanding the finiteness properties of groups is to study
how difficult it is to detect if loops in a \(K(G,1)\) of a group \(G\) are null-homotopic. For loops, this is measured by the Dehn function \(\delta _{G}(h)\) of the group \(G\) which is
defined as the maximal area that a minimal filling disc of a loop of length at most \(n\) can have.
The connection between finiteness properties of groups and the Dehn functions are well known.
For example, a finitely presented group has solvable word problem if and only if the Dehn function for a finite presentation of this group is recursive.
Many efforts have been made into the exact determination of Dehn functions on subgroups which appear as kernels of homomorphisms from direct products of groups on infinite cyclic groups. More precisely, if \(\Gamma _{1},\Gamma _{2},\Gamma _{3}\) are simplicial graphs and \(A_{\Gamma _{1}},A_{\Gamma _{2}},A_{\Gamma _{3}}\) are the corresponding right-angled Artin groups, then in [\textit{W. Carter} and \textit{M. Forester}, Math. Ann. 368, No. 1--2, 671--683 (2017; Zbl 1423.20042)] it is proved that the kernel of the epimorphism \(\varphi : A_{\Gamma _{1}}\times A_{\Gamma _{2}}\times A_{\Gamma _{3}} \longrightarrow \mathbb{Z}\) sending each standard generator to 1, namely the Bestvina-Brady group \(BB_{\Gamma}\) has quadratic Dehn function. Already many results concerning the Dehn functions of the Stallings-Bieri groups \(SB_{k}\), namely the kernels of the epimorphisms from products of \(k\) free groups onto \(\mathbb{Z}\) (\(F_{2}\times F_{2}\times \cdots \times F_{2} \longrightarrow \mathbb{Z}\)), are obtained. The groups \(SB_{k}\) are the first examples of type
\(\mathcal{F}_{k-1}\) and not \(\mathcal{F}_{k}\) for any \(k\geq 3\).
These results raise the question if one can also understand the Dehn functions of more general subgroups \(K\leq G_{1}\times \cdots \times G_{n}\) of direct products of groups and in
particular of coabelian subgroups (of corank \(l\)) arising as kernel of a surjective homomorphism
\(\varphi : G_{1}\times \cdots \times G_{n}\longrightarrow \mathbb{Z}^{l}\).
In the present paper, the authors develop new methods for computing the precise Dehn functions of coabelian subgroups of direct products of groups in terms of the Dehn functions of the factors.
To obtain them, they translate the main result of [\textit{W. Carter} and \textit{M. Forester}, Math. Ann. 368, No. 1--2, 671--683 (2017; Zbl 1423.20042)] to an algebraic setting and then generalise it in two different ways.
This formulation in algebraic terms allows to prove results for arbitrary groups that do not
need to admit an action on a simply connected cube complex.
Their main results are:
Theorem A (Theorem 3.2 in the paper) For \(1\leq i\leq 3\), let \(G_{i}\) be finitely presented groups and let
\(1 \longrightarrow N_{i} \longrightarrow G_{i} \stackrel{\varphi _{i}}{\longrightarrow} \mathbb{Z}^{m} \longrightarrow 1\) be right-split short exact sequences. Let \(\varphi : G_{1}\times G_{2}\times G_{3} \longrightarrow \mathbb{Z}^{m}\) be defined by \(\varphi (g_{1}, g_{2}, g_{3}) = \sum_{i = 1}^{3}\varphi _{i}(g_{i})\).
Let \(\bar{f}\) be the superadditive closure of the Dehn function \(f\) of \(G_{1}\times G_{2}\times G_{3}\). Then
\(K :=\ker(\varphi )\) is finitely presented and its Dehn function satisfies \(f(n)\preceq \delta_{K}(n)\preceq \bar{f}(n)\cdot \log(n)\). If, moreover, \(\frac{f(n)}{n}\) is superadditive, then
\(f(n)\asymp \delta _{K}(n)\).
Theorem B (Theorem 4.2 in the paper) Let \(G_{1}, \dots ,G_{4}\) be finitely presented groups and let \(1 \longrightarrow N_{i} \longrightarrow G_{i} \stackrel{\varphi _{i}}{\longrightarrow} \mathbb{Z}^{m}\longrightarrow 1\) be short
exact sequences. Suppose that there is a \(P\) such that each of these sequences \(P\)-splits.
Let \(\varphi : G_{1}\times G_{2}\times G_{3}\times G_{4} \longrightarrow \mathbb{Z}^{m}\) be defined by
\(\varphi (g_{1},g_{2},g_{3})=\sum _{i=1}^{4}\varphi _{i}(g_{i})\) and let \(f(n)\) be the Dehn function
of \(G_{1}\times G_{2}\times G_{3}\times G_{4}\). Then
\(K :=\ker(\varphi )\) is finitely presented and its Dehn function satisfies \(f(n)\preceq \delta_{K}(n)\preceq \bar{f}(n)\cdot \log(n)\). If, moreover, \(\frac{f(n)}{n}\) is superadditive, then
\(f(n)\asymp \delta _{K}(n)\).
Here a function \(f : \mathbb{N} \longrightarrow \mathbb{R}_{\geq 0}\) is called superadditive if \(f(n+m)\geq f(m)+f(n)\). The superadditive closure \(\bar{f}\) of \(f\) is the smallest superadditive function with \(\bar{f}(n) \geq f(n)\).
For monotonely increasing functions \(f, g : \mathbb{R}_{>0} \longrightarrow \mathbb{R}\), \(f\preceq g\) means that there is a constant \(C>0\) such that \(f(n)\leq Cg(Cn+C)+Cn+C\) for all \(n\in \mathbb{R}_{>0}\) and \(f\asymp g\) means that \(f \preceq g \preceq f\).
Also, the pair \(P = \{ A, B \}\) is said be a factoring of \(\mathbb{Z}^{m}\) if \(A, B\leq \mathbb{Z}^{m}\) and \(\mathbb{Z}^{m} = A\oplus B\). For a given factoring \(P\), the short exact sequence \(1 \longrightarrow N \longrightarrow G \stackrel{\varphi }{\longrightarrow} \mathbb{Z}^{m} \longrightarrow 1\) \(P\)-splits, if there are maps \(s^{1} : A \longrightarrow G\), \(s^{2} : B \longrightarrow G\)
such that \(\varphi \circ s^{i} = id\).
The proofs provide a constructive way of filling a loop with a disk.
Several applications of these methods are given.
Here we quote some of them.
Theorem C (Theorem 5.4 in the paper). Let \(r\geq 3\), let \(G_{1}, \dots, G_{r}\) be finitely presented groups and let \(K :=\ker (G_{1}\times \cdots \times G_{r}) \stackrel{\varphi }{\longrightarrow} \mathbb{Z}^{l}\) be a coabelian subgroup of corank \(l\geq 0\). Denote by \(f\) the Dehn function of
\(G_{1}\times \cdots \times G_{r}\). If \(\lceil \frac{l}{2} \rceil \leq \frac{r}{4}\) and the restriction
of \(\varphi\) to every factor is virtually surjective, then \(f(n)\preceq \delta_{K}(n)\preceq \bar{f}(n)\cdot \log(n)\). If, moreover, \(\frac{f(n)}{n}\) is superadditive, then
\(\delta _{K}(n)\asymp f(n) \).
This result improves a result of \textit{W. Dison} [Isoperimetric functions for subdirect products and Bestvina-Brady groups. London: Imperial College (PhD Thesis) (2008), Theorem 11.3 (4)].
Theorem D (Theorem 5.5 in the paper). Let \(f(n)\geq n^{2}\) be a function which can be realised as Dehn function of a group \(G\) of type \(\mathcal{F}_{\infty}\) and let \(k\geq 3\). Then there
is a 1-ended irreducible group \(K\) of type \(\mathcal{F}_{k-1}\) and not \(\mathcal{F}_{k}\)
whose Dehn function satisfies \(\bar{f}(n)\preceq \delta_{K}(n)\preceq \bar{f}(n)\cdot \log(n)\). If, moreover, \(\frac{f(n)}{n}\) is superadditive, then
\(\delta _{K}(n)\asymp f(n) \).
An infinite group \(G\) is called irreducible, if it does not have a finite index subgroup of the form
\(H_{1}\times H_{2}\) with \(H_{1}, H_{2}\) infinite.
Theorem E (Theorem 5.6 in the paper). Let \(A_{\Gamma } = H_{1}\times H_{2}\times H_{3}\times H_{4}\) be a product of four right-angled Artin groups. Suppose that we have \(P\)-split maps
\(\varphi _{i}: H_{i} \longrightarrow \mathbb{Z}^{m}\). Let \(\varphi = \sum \varphi _{i}\). Then \(K =\ker (\varphi )\) has quadratic Dehn function.
Theorem F (Theorem 5.8 in the paper). Let \(A_{\Gamma } = H_{1}\times H_{2}\times H_{3}\) be a product of three right-angled Artin groups. Suppose that we have split surgections
\(\varphi _{i} : H_{i} \longrightarrow \mathbb{Z}^{m}\). Let \(\varphi = \sum \varphi _{i}\). Then \(K =\ker (\varphi )\) has quadratic Dehn function.
Compare with Corollary 4.3 in [\textit{W. Carter} and \textit{M. Forester}, Math. Ann. 368, No. 1--2, 671--683 (2017; Zbl 1423.20042)].
Reviewer: Dimitrios Varsos (Athína)Some new CAT(0) free-by-cyclic groupshttps://zbmath.org/1526.200622024-02-15T19:53:11.284213Z"Lyman, Rylee Alanza"https://zbmath.org/authors/?q=ai:lyman.rylee-alanzaAuthor's abstract: We show the existence of several new infinite families of polynomially-growing automorphisms of free groups whose mapping tori are CAT(0) free-by-cyclic groups. Such mapping tori are thick, and thus not relatively hyperbolic. These are the first families comprising infinitely many examples for each rank of the nonabelian free group; they contrast strongly with Gersten's example of a thick free-by-cyclic group which cannot be a subgroup of a CAT(0) group.
Reviewer: Alexander Felshtyn (Szczecin)Normal subgroups of SimpHAtic groupshttps://zbmath.org/1526.200642024-02-15T19:53:11.284213Z"Osajda, Damian"https://zbmath.org/authors/?q=ai:osajda.damian-lIn the paper under review, the author studies the groups that act geometrically on a simply connected simplicially hereditarily aspherical complex (SimpHAtic-groups and SimpHAtic-complexes, for brevity). He shows that finitely presented normal subgroups of the SimpHAtic groups are either: finite, or of finite index, or virtually free. This result applies, in particular, to normal subgroups of systolic groups.
He also shows that non-uniform lattices in SimpHAtic complexes are not finitely presentable and that finitely presented groups acting properly on such complexes act geometrically on SimpHAtic complexes.
Reviewer: Egle Bettio (Venezia)Equivariant Morse theory on Vietoris-Rips complexes and universal spaces for proper actionshttps://zbmath.org/1526.200662024-02-15T19:53:11.284213Z"Varisco, Marco"https://zbmath.org/authors/?q=ai:varisco.marco"Zaremsky, Matthew C. B."https://zbmath.org/authors/?q=ai:zaremsky.matthew-curtis-burkholderA \(G\)-CW complex is a CW complex with a \(G\)-action such that, for all open cells \(e\) and all \(g\in G\), if \(g\cdot e\cap e\,\not =\,\emptyset\), then \(g\cdot x\,=\,x\) for all \(x\in e\). A \(G\)-CW complex is proper if and only if all point stabilizers are finite. A \(G\)-CW complex is called finite if it is cocompact, or, equivalently, if it only has finitely many \(G\)-orbits of cells.
A universal space for proper actions of a discrete group \(G\) is a proper \(G\)-CW complex \(\underline{E}G\) such that, for each finite subgroup \(H\) of \(G\), the \(H\)-fixed point set \((\underline{E}G)^{H}\) is contractible. Universal spaces for proper actions exist, for any group \(G\) and are unique up to \(G\)-homotopy. A central question is whether a given group \(G\) has universal space for proper actions that is finite. In [\textit{D. Meintrup} and \textit{T. Schick}, New York J. Math. 8, 1--7 (2002; Zbl 0990.20027)] it is proved that hyperbolic groups have finite universal spaces for proper actions. Also, by a result of \textit{P. Ontaneda} [Topology 44, No. 1, 47--62 (2005; Zbl 1068.53026)], it follows that CAT(0) groups have finite universal spaces for proper actions.
A natural simultaneous generalization of hyperbolic and CAT(0) groups is
given by asymptotically CAT(0) groups, which were introduced and studied by \textit{A. Kar} [Publ. Mat., Barc. 55, No. 1, 67--91 (2011; Zbl 1271.20057)] (see Definition 1.2 in the paper).
The class of asymptotically CAT(0) groups contains all hyperbolic groups and all CAT(0) groups and is closed under taking finite products, amalgamated free products over finite subgroups, HNN extensions along finite subgroups and relatively hyperbolic overgroups.
The main result of the authors is:
{Theorem.} (Theorem 3.5 in the paper) Let \(G\) be a group acting properly and cocompactly by isometries on an asymptotically CAT(0) geodesic metric space \(X\). Consider the orbit \(G\cdot x_{0}\) of an arbitrary point \(x_{0}\in X\) with the induced metric from \(X\). Then, for any sufficiently large \(t\in \mathbb{R}\), the Vietoris-Rips complex \(\mathcal{VR}_{t}(G\cdot x_{0})\) is a finite model for the universal space for proper actions \(\underline{E}G\).
{Corollary.} (Theorem 1.1 in the paper) All asymptotically CAT(0) groups have finite universal spaces for proper actions.
The proof of this result relies on an equivariant version of \textit{M. Bestvina} and \textit{N. Brady} discrete Morse theory (see in [Invent. Math. 129, No. 3, 445--470 (1997; Zbl 0888.20021)]). This development needs some care to formulate the appropriate equivariant definitions and statements and in checking the equivariance of the constructions. As the authors pointed out this is of independent interest. This approach here relies heavily on the ideas from [\textit{M. C. B. Zaremsky}, Am. J. Math. 144, No. 5, 1177--1200 (2022; Zbl 1504.57043)], but is completely independent from the results proved there.
This equivariant version is applied to Vietoris-Rips complexes. The definition of a Vietoris-Rips complex that is adopted here is not the typical definition. The typical definition yields a simplicial complex whose barycentric subdivision equals ours and so the two definitions are naturally homeomorphic. Here, the definition is formulated in terms of posets and the language and tools from the world of posets turn out to be convenient for the arguments to prove the main result.
Two key results are proved before (Propositions 2.1 and 3.4) which lead to the main result.
The paper concludes with two interesting questions (see the relevant discussion at the end of the paper).
1. Is there a natural condition weaker than asymptotically CAT(0) for which the proof of the theorem above still works? It would also be very interesting to find a condition similar to CAT(0) or asymptotically CAT(0) that is invariant under quasi-isometries and for which the result of the theorem above is true.
2. Is it true or false that if \(G\) is an asymptotically CAT(0) or even a CAT(0) group, then its own Vietoris-Rips complex \(\mathcal{VR}_{t}(G)\) with respect to a word metric is
an \(\underline{E}G\) for large enough \(t\)?
Reviewer: Dimitrios Varsos (Athína)Dominating surface-group representations into \(\mathrm{PSL}_2 (\mathbb{C})\) in the relative representation varietyhttps://zbmath.org/1526.300592024-02-15T19:53:11.284213Z"Gupta, Subhojoy"https://zbmath.org/authors/?q=ai:gupta.subhojoy"Su, Weixu"https://zbmath.org/authors/?q=ai:su.weixuSummary: Let \(\rho\) be a representation of the fundamental group of a punctured surface into \({\mathrm{PSL}_2 (\mathbb{C})}\) that is not Fuchsian. We prove that there exists a Fuchsian representation that strictly dominates \(\rho\) in the simple length spectrum, and preserves the boundary lengths. This extends a result of \textit{F. Guéritaud} et al. [Pac. J. Math. 275, No. 2, 325-359 (2015; Zbl 1346.57005)] to the case of \(\mathrm{PSL}_2 (\mathbb{C})\)-representations. Our proof involves straightening the pleated plane in \(\mathbb{H}^3\) determined by the Fock-Goncharov coordinates of a framed representation, and applying strip-deformations.Yamada polynomial and associated link of \(\theta \)-curveshttps://zbmath.org/1526.570082024-02-15T19:53:11.284213Z"Huh, Youngsik"https://zbmath.org/authors/?q=ai:huh.youngsikThe discovery of polynomial invariants for knots and links, sparked by V. F. R. Jones, has led to the development of polynomial invariants for spatial graphs. One such invariant is the Yamada polynomial, which is often used to distinguish between spatial graphs in practical applications. Specifically, when it comes to \(\theta\)-curves, this polynomial serves as an ambient isotopy invariant after normalization.
On the other hand, to each \(\theta\)-curve, a 3-component link can be associated as an ambient isotopy invariant. The advantage of these associated links is that the invariants of these links can be applied as invariants for \(\theta\)-curves.
In this paper, the author explores the relationship between the normalized Yamada polynomial of \(\theta\)-curves and the Jones polynomial of their associated 3-component links. He demonstrates that, as a corollary, these two polynomials are equivalent for Brunnian \(\theta\)-curves. To achieve this, the Jaeger polynomial of the spatial graphs is examined, with a particular specialization being equivalent to the Yamada polynomial.
Reviewer: Qingying Deng (Xiangtan)Heegaard genus, degree-one maps, and amalgamation of 3-manifoldshttps://zbmath.org/1526.570142024-02-15T19:53:11.284213Z"Li, Tao"https://zbmath.org/authors/?q=ai:li.tao.18|li.tao|li.tao.2|li.tao.9|li.tao.7|li.tao.4|li.tao.1|li.tao.3|li.tao.11|li.tao.10Degree-one maps are an important topic in low dimensional topology and play an important role in the study of \(3\)-manifolds. It has been known for a long time that maps of nonzero degree between surfaces are standard. However, many important questions remain open for maps between 3-manifolds. One of the most fundamental questions on degree-one maps between 3-manifolds is the relation between their Heegaard genera as follows:
\textbf{Conjecture 1.1.} Let \(M\) and \(N\) be closed orientable \(3\)-manifolds and suppose there is a degree-one map \(f: M\rightarrow N\). Then \(g(M)\geq g(N)\), where \(g(M)\) is the Heegaard genus of \(M\).
Also, we can formulate the question in Conjecture 1.1 as a question about surgery as follows:
\textbf{Conjecture 1.2.} Suppose \(M=W\cup_{T}V\) with \(T=\partial W=\partial V=W\cap V\) a genus-\(g\) surface. Suppose \(W\) satisfies the two conditions above. Let \(N\) be the closed \(3\)-manifold obtained by replacing \(W\) with a genus-\(g\) handlebody \(H\) such that each \(\gamma_{i}\) in the conditions above bounds a disk in \(H\). Then \(g(M)\geq g(N)\).
In the paper under review, the author proves that Conjecture 1.2 holds if \(W\) is a knot exterior in a homology sphere. Also, he gives some useful corollaries.
Reviewer: Kun Du (Lanzhou)Concordance of spatial graphshttps://zbmath.org/1526.570172024-02-15T19:53:11.284213Z"Lappo, Egor"https://zbmath.org/authors/?q=ai:lappo.egorSummary: We define smooth notions of concordance and sliceness for spatial graphs. We prove that sliceness of a spatial graph is equivalent to a condition on a set of linking numbers together with sliceness of a link associated with the graph. This generalizes the result of \textit{K. Taniyama} [Math. Proc. Camb. Philos. Soc. 113, No. 1, 97--106 (1993; Zbl 0783.57003)] for \(\theta\)-curves.Self-dual maps. II: Links and symmetryhttps://zbmath.org/1526.570182024-02-15T19:53:11.284213Z"Montejano, Luis"https://zbmath.org/authors/?q=ai:montejano.luis-pedro|montejano.luis"Ramírez Alfonsín, Jorge L."https://zbmath.org/authors/?q=ai:ramirez-alfonsin.jorge-luis"Rasskin, Iván"https://zbmath.org/authors/?q=ai:rasskin.ivanSummary: In this paper, we investigate representations of links that are either \textit{centrally symmetric} in \(\mathbb{R}^3\) or \textit{antipodally symmetric} in \(\mathbb{S}^3\). By using the notions of \textit{antipodally self-dual} and \textit{antipodally symmetric} maps, introduced and studied by the authors in [\textit{L. Montejano} et al., SIAM J. Discrete Math. 36, No. 3, 1551--1566 (2022; Zbl 1492.05149)], we are able to present sufficient combinatorial conditions for a link \(L\) to admit such representations. The latter naturally provide sufficient conditions for \(L\) to be \textit{amphichiral}. We also introduce another (closely related) method yielding again sufficient conditions for \(L\) to be amphichiral. We finally prove that a link \(L\), associated to a map \(G\), is amphichiral if the \textit{self-dual pairing} of \(G\) is not one of 6 specific cases among the classification of the \textit{24 self-dual pairing} \(\mathrm{Cor}(G)\rhd\mathrm{Aut}(G)\).Orientable and non-orientable regular maps with given exponent grouphttps://zbmath.org/1526.570192024-02-15T19:53:11.284213Z"Asciak, Kirstie"https://zbmath.org/authors/?q=ai:asciak.kirstie"Conder, Marston D. E."https://zbmath.org/authors/?q=ai:conder.marston-d-e"Pavlíková, Soňa"https://zbmath.org/authors/?q=ai:pavlikova.sona"Širáň, Jozef"https://zbmath.org/authors/?q=ai:siran.jozefSummary: With the help of the parallel product (also known as the join) of maps given by subgroups of triangle groups, and some facts about automorphisms of products of simple groups, we extend a 2016 theorem of \textit{M. Conder} and \textit{J. Širáň} [Bull. Lond. Math. Soc. 48, No. 6, 1013--1017 (2016; Zbl 1372.57036)] on exponent groups of orientable maps, by proving that for every \(d \geq 3\) and every group \(U\) of units\(\mod d\) containing \(-1\), there exist infinitely many non-orientable regular maps of valency \(d\) with exponent group equal to \(U\).