Mapping class groups have a unique Polish group structure
We prove that mapping class groups of surfaces and of locally finite connected graphs support a unique Polish group structure.
math.GT↗
arXiv subjects
Sumun Iyer
Publications and source records attributed to Sumun Iyer.
Universal minimal flows of homeomorphism groups of continua
We define a combinatorial property of a projective Fraisse category which we call the \emph{approximate Ramsey property}. Let $F$ be a continuum, $G$ a closed subgroup of the homeomorphism group of $F$, and $\mathbb{F}$ the limit of projective Fraisse category $\mathcal{F}$ such that $\textrm{Aut}(\mathbb{F})$ is dense in $G$. We prove that $\mathcal{F}$ has the approximate Ramsey property if and only if $G$ is extremely amenable. We prove that the group of homeomorphisms of the universal pseudo-solenoid has non-metrizable universal minimal flow.
math.LO↗
Asymptotic dimension and hyperfiniteness of generic Cantor actions
We show that for a countable discrete group which is locally of finite asymptotic dimension, the generic continuous action on Cantor space has hyperfinite orbit equivalence relation. In particular, this holds for free groups, answering a question of Frisch-Kechris-Shinko-Vidny\'anszky.
math.LO↗
A Ramsey-type phenomenon in two and three dimensional simplices
We develop a Ramsey-like theorem for subsets of the two and three-dimensional simplex. A generalization of the combinatorial theorem presented here to all dimensions would produce a new proof that $\textrm{Homeo}_+[0,1]$ is extremely amenable (a theorem due to Pestov) using general results of Uspenskij on extreme amenability in homeomorphism groups.
math.CO↗
Direct limits of large orbits and the Knaster continuum homeomorphism group
The main result is that the group $\textrm{Homeo} (K)$ of homeomorphisms of the universal Knaster continuum contains an open subgroup with a comeager conjugacy class. Actually, this open subgroup is the very natural subgroup consisting of degree-one homeomorphisms. We give a general fact about finding comeager orbits in Polish group actions which are approximated densely by direct limits of actions with comeager orbits. The main theorem comes as a result of this fact and some finer analysis of the conjugacy action of the group $\textrm{Homeo}_+[0,1]$.
math.DS↗
The homeomorphism group of the universal Knaster continuum
We define a projective Fraissé family whose limit approximates the universal Knaster continuum. The family is such that the group $\textrm{Aut}(\mathbb{K})$ of automorphisms of the Fraissé limit is a dense subgroup of the group, $\textrm{Homeo}(K)$, of homeomorphisms of the universal Knaster continuum. We prove that both $\textrm{Aut}(\mathbb{K})$ and $\textrm{Homeo}(K)$ have universal minimal flow homeomorphic to the universal minimal flow of the free abelian group on countably many generators. The computation involves proving that both groups contain an open, normal subgroup which is extremely amenable.
math.LO↗
Tiling Billards on Triangle Tilings, and Interval Exchange Transformations
We consider the dynamics of light rays in triangle tilings where triangles are transparent and adjacent triangles have equal but opposite indices of refraction. We find that the behavior of a trajectory on a triangle tiling is described by an orientation-reversing three-interval exchange transformation on the circle, and that the behavior of all the trajectories on a given triangle tiling is described by a polygon exchange transformation. We show that, for a particular choice of triangle tiling, certain trajectories approach the Rauzy fractal, under rescaling.
math.DS↗
Domination and Upper Domination of Direct Product Graphs
The unitary Cayley graph of $\mathbb{Z} /n \mathbb{Z}$, denoted $X_{\mathbb{Z} / n \mathbb{Z}}$, has vertices $0,1, \dots, n-1$ with $x$ adjacent to $y$ if $x-y$ is relatively prime to $n$. We present results on the tightness of the known inequality $γ(X_{\mathbb{Z} / n \mathbb{Z}})\leq γ_t(X_{\mathbb{Z} / n \mathbb{Z}})\leq g(n)$, where $γ$ and $γ_t$ denote the domination number and total domination number, respectively, and $g$ is the arithmetic function known as Jacobsthal's function. In particular, we construct integers $n$ with arbitrarily many distinct prime factors such that $γ(X_{\mathbb{Z} / n \mathbb{Z}})\leqγ_t(X_{\mathbb{Z} / n \mathbb{Z}})\leq g(n)-1$. Extending work of Mekiš, we give lower bounds for the domination numbers of direct products of complete graphs. We also present a simple conjecture for the exact values of the upper domination numbers of direct products of balanced, complete multipartite graphs and prove the conjecture in certain cases. We end with some open problems.
math.CO↗
Star coloring splitting graphs of cycles
A star coloring of a graph $G$ is a proper vertex coloring such that the subgraph induced by any pair of color classes is a star forest. The star chromatic number of $G$ is the minimum number of colors needed to star color $G$. In this paper we determine the star-chromatic number of the splitting graphs of cycles of length $n$ with $n \equiv 1 \pmod 3$ and $n=5$, resolving an open question of Furnmańczyk, Kowsalya, and Vernold Vivin.
math.CO↗