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Peter Wong

Publications and source records attributed to Peter Wong.

At least 19 recordsLinked to original sources
arXiv2026

Fixed point free homeomorphisms and the $R_{\infty}$-property

Let $f:M\to M$ be a diffeomorphism on a compact connected smooth manifold of dimension at least $5$. It is known that the Nielsen number $N(f)$ of $f$ vanishes if and only if $f$ is isotopic to a fixed point free map. For any $n\ge 5$, there exists a compact $n$-dimensional nilmanifold $M$ such that {\it every} self homeomorphism on $M$ is isotopic to a fixed point free map. The proof depends on the fact that nilmanifolds are of {\it Jiang-type} and the fundamental group $\pi_1(M)$ has the property $R_{\infty}$ for certain nilmanifolds $M$. In this paper we construct the first known examples (an infinite family) of non-aspherical closed manifolds whose fundamental groups have property $R_{\infty}$ and that are also of Jiang-type. In particular, every self homeomorphism on such a manifold is isotopic to be fixed point free. The main objective of this work is to construct non-aspherical manifolds whose fundamental groups have property $R_{\infty}$ and that are also of Jiang-type.

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arXiv2025

Property $R_{\infty}$ for groups with infinitely many ends

We show that an accessible group with infinitely many ends has property $R_{\infty}$. That is, it has infinitely many twisted conjugacy classes for any twisting automorphism. We deduce that having property $R_{\infty}$ is undecidable amongst finitely presented groups. We also show that the same is true for a wide class of relatively hyperbolic groups, filling in some of the gaps in the literature. Specifically, we show that a non-elementary, finitely presented relatively hyperbolic group with finitely generated peripheral subgroups which are not themselves relatively hyperbolic, has property $R_{\infty}$.

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arXiv2021

The BNS invariants of the generalized solvable Baumslag-Solitar groups and of their finite index subgroups

We compute the Bieri-Neumann-Strebel invariants $\Sigma^1$ for the generalized solvable Baumslag-Solitar groups $\Gamma_n$ and their finite index subgroups. Using $\Sigma^1$, we show that certain finite index subgroups of $\Gamma_n$ cannot be isomorphic to $\Gamma_{k}$ for any $k$. In addition, we use the BNS-invariants to give a new proof of property $R_\infty$ for the groups $\Gamma_n$ and their finite index subgroups.

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arXiv2021

Twisted conjugacy and commensurability invariance

A group $G$ is said to have property $R_{\infty}$ if for every automorphism $φ\in {\rm Aut}(G)$, the cardinality of the set of $φ$-twisted conjugacy classes is infinite. Many classes of groups are known to have such property. However, very few examples are known for which $R_{\infty}$ is {\it geometric}, i.e., if $G$ has property $R_{\infty}$ then any group quasi-isometric to $G$ also has property $R_{\infty}$. In this paper, we give examples of groups and conditions under which $R_{\infty}$ is preserved under commensurability. The main tool is to employ the Bieri-Neumann-Strebel invariant.

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arXiv2020

Twisted conjugacy in free products

Let $ϕ:G\to G$ be an automorphism of a group which is a free-product of finitely many groups each of which is freely indecomposable and two of the factors contain proper finite index characteristic subgroups. We show that $G$ has infinitely many $ϕ$-twisted conjugacy classes. As an application, we show that if $G$ is the fundamental group of a three-manifold that is not irreducible, then $G$ has property $R_\infty$, that is, there are infinitely many $ϕ$-twisted conjugacy classes in $G$ for every automorphism $ϕ$ of $G$.

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arXiv2020

Computation of Nielsen and Reidemeister coincidence numbers for multiple maps

Let $f_1,...,f_k:M\to N$ be maps between closed manifolds, $N(f_1,...,f_k)$ and $R(f_1,...,f_k)$ be the Nielsen and the Reideimeister coincidence numbers respectively. In this note, we relate $R(f_1,...,f_k)$ with $R(f_1,f_2),...,R(f_1,f_k)$. When $N$ is a torus or a nilmanifold, we compute $R(f_1,...,f_k)$ which, in these cases, is equal to $N(f_1,...,f_k)$.

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arXiv2018

Exponents of $[Ω(\mathbb S^{r+1}), Ω(Y)]$

We investigate the exponents of the total Cohen groups $[Ω(\mathbb S^{r+1}), Ω(Y)]$ for any $r\ge 1$. In particular, we show that for $p\ge 3$, the $p$-primary exponents of $[Ω(\mathbb S^{r+1}), Ω(\mathbb S^{2n+1})]$ and $[Ω(\mathbb S^{r+1}), Ω(\mathbb S^{2n})]$ coincide with the $p$-primary homotopy exponents of spheres $\mathbb S^{2n+1}$ and $\mathbb S^{2n}$, respectively. We further study the exponent problem when $Y$ is a space with the homotopy type of $Σ(n)/G$ for a homotopy $n$-sphere $Σ(n)$, the complex projective space $\mathbb{C}P^n$ for $n\ge 1$ or the quaternionic projective space $\mathbb{H}P^n$ for $1\le n\le \infty$.

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arXiv2018

Coincidence Wecken property for nilmanifolds

Let $f,g:X\to Y$ be maps from a compact infra-nilmanifold $X$ to a compact nilmanifold $Y$ with $\dim X\ge \dim Y$. In this note, we show that a certain Wecken type property holds, i.e., if the Nielsen number $N(f,g)$ vanishes then $f$ and $g$ are deformable to be coincidence free. We also show that if $X$ is a connected finite complex $X$ and the Reidemeister coincidence number $R(f,g)=\infty$ then $f\sim f'$ so that $C(f',g)=\{x\in X \mid f'(x)=g(x)\}$ is empty.

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arXiv2017

Mapping degrees between spherical $3$-manifolds

Let $D(M,N)$ be the set of integers that can be realized as the degree of a map between two closed connected orientable manifolds $M$ and $N$ of the same dimension. For closed $3$-manifolds with $S^3$-geometry $M$ and $N$, every such degree $deg f\equiv \overline{deg}ψ$ $(|π_1(N)|)$ where $0\le \overline{deg}ψ<|π_1(N)|$ and $\overline{deg}ψ$ only depends on the induced homomorphism $ψ=f_π$ on the fundamental group. In this paper, we calculate explicitly the set $\{\overline{deg}ψ\}$ when $ψ$ is surjective and then we show how to determine $\overline{deg}(ψ)$ for arbitrary homomorphisms. This leads to the determination of the set $D(M,N)$.

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arXiv2017

Obstruction theory for coincidences of multiple maps

Let $f_1,..., f_k:X\to N$ be maps from a complex $X$ to a compact manifold $N$, $k\ge 2$. In previous works \cite{BLM,MS}, a Lefschetz type theorem was established so that the non-vanishing of a Lefschetz type coincidence class $L(f_1,...,f_k)$ implies the existence of a coincidence $x\in X$ such that $f_1(x)=...=f_k(x)$. In this paper, we investigate the converse of the Lefschetz coincidence theorem for multiple maps. In particular, we study the obstruction to deforming the maps $f_1,...,f_k$ to be coincidence free. We construct an example of two maps $f_1,f_2:M\to T$ from a sympletic $4$-manifold $M$ to the $2$-torus $T$ such that $f_1$ and $f_2$ cannot be homotopic to coincidence free maps but for {\it any} $f:M\to T$, the maps $f_1,f_2,f$ are deformable to be coincidence free.

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arXiv2017

Fixed point sets of equivariant fiber-preserving maps

Given a selfmap $f:X\to X$ on a compact connected polyhedron $X$, H. Schirmer gave necessary and sufficient conditions for a nonempty closed subset $A$ to be the fixed point set of a map in the homotopy class of $f$. R. Brown and C. Soderlund extended Schirmer's result to the category of fiber bundles and fiber-preserving maps. The objective of this paper is to prove an equivariant analogue of Brown-Soderlund theorem result in the category of $G$-spaces and $G$-maps where $G$ is a finite group.

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arXiv2015

Automorphisms of Higher Rank Lamplighter Groups

Let $Γ_d(q)$ denote the group whose Cayley graph with respect to a particular generating set is the Diestel-Leader graph $DL_d(q)$, as described by Bartholdi, Neuhauser and Woess. We compute both $Aut(Γ_d(q))$ and $Out(Γ_d(q))$ for $d \geq 2$, and apply our results to count twisted conjugacy classes in these groups when $d \geq 3$. Specifically, we show that when $d \geq 3$, the groups $Γ_d(q)$ have property $R_{\infty}$, that is, every automorphism has an infinite number of twisted conjugacy classes. In contrast, when $d=2$ the lamplighter groups $Γ_2(q)=L_q = {\mathbb Z}_q \wr {\mathbb Z}$ have property $R_{\infty}$ if and only if $(q,6) \neq 1$.

math.GR↗
arXiv2014

On the group structure of $[Ω\mathbb S^2, ΩY]$

Let $J(X)$ denote the James construction on a space $X$ and $J_n(X)$ be the $n$-th stage of the James filtration of $J(X)$. It is known that $[J(X),ΩY]\cong \lim\limits_{\leftarrow} [J_n(X),ΩY]$ for any space $Y$. When $X=\mathbb S^1$, the circle, $J(\mathbb S^1)=ΩΣ\mathbb S^1=Ω\mathbb S^2$. Furthermore, there is a bijection between $[J(\mathbb S^1),ΩY]$ and the product $\prod_{i=2}^\infty π_i(Y)$, as sets. In this paper, we describe the group structure of $[J_n(\mathbb S^1),ΩY]$ by determining the co-multiplication structure on the suspension $ΣJ_n(\mathbb S^1)$.

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arXiv2012

The Geometric Invariants of Group Extensions

In this paper, we compute the Σ^n(G) and Ω^n(G) invariants when 1 \rightarrow H \rightarrow G \rightarrow K \rightarrow 1 is a short exact sequence of finitely generated groups with K finite. We also give sufficient conditions for G to have the R_{\infty} property in terms of Ω^n(H) and Ω^n(K) when either K is finite or the sequence splits. As an application, we construct a group F \rtimes? Z_2 where F is the R. Thompson's group F and show that F \rtimes Z_2 has the R_{\infty} property while F is not characteristic.

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arXiv2011

The Geometric Invariants of Group Extensions Part I: Finite Extensions

In this note, we compute the Σ^1(G) invariant when 1 {\to} H {\to} G {\to} K {\to} 1 is a short exact sequence of finitely generated groups with K finite. As an application, we construct a group F semidirect Z_2 where F is the R. Thompson's group F and show that F semidirect Z_2 has the R-infinity property while F is not characteristic. Furthermore, we construct a finite extension G with finitely generated commutator subgroup G' but has a finite index normal subgroup H with infinitely generated H'.

math.GR↗
arXiv2011

The Geometric Invariants of Group Extensions Part II: Split Extensions

We compute the Ω^1(G) invariant when 1 {\to} H {\to} G {\to} K {\to} 1 is a split short exact sequence. We use this result to compute the invariant for pure and full braid groups on compact surfaces. Applications to twisted conjugacy classes and to finite generation of commutator subgroups are also discussed.

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