arXiv subjects

Ryosuke Mineyama

Publications and source records attributed to Ryosuke Mineyama.

5 recordsLinked to original sources

arXiv2019

Fibered commensurability on $\mathrm{Out}(F_{n})$

We define and discuss a notion called fibered commensurability of outer automorphisms of free groups. This notion lets us study symmetry of outer automorphisms. The notion of fibered commensurability is first defined by Calegari-Sun-Wang on mapping class groups. The Nielsen-Thurston type of mapping classes is a commensurability invariant. One of the important facts of fibered commensurability on mapping class groups is for the case of pseudo-Anosovs, there is a unique minimal element in each fibered commensurability class. For outer automorphisms, we first show that being atoroidal and fully irreducible is a commensurability invariant. Then for such outer automorphisms, we prove that there is a unique minimal element in each fibered commensurability class, under a certain asymmetry condition on the ideal Whitehead graphs.

math.GT↗

arXiv2017

Distribution of accumulation points of roots for type $(n-1,1)$ Coxeter groups

In this paper, we investigate the set of accumulation points of normalized roots of infinite Coxeter groups for certain class of their action. Concretely, we prove the conjecture proposed in [6, Section 3.2] in the case where the equipped Coxeter matrices are of type $(n-1,1)$, where $n$ is the rank. Moreover, we obtain that the set of such accumulation points coincides with the closure of the orbit of one point of normalized limit roots. In addition, in order to prove our main results, we also investigate some properties on fixed points of the action.

math.GR↗

arXiv2017

On coarse geometric aspects of the Hilbert geometry

We begin a coarse geometric study of Hilbert geometry. Actually we give a necessary and sufficient condition for the natural boundary of a Hilbert geometry to be a corona, which is a nice boundary in coarse geometry. In addition, we show that any Hilbert geometry is uniformly contractible and with coarse bounded geometry. As a consequence of these we see that the coarse Novikov conjecture holds for a Hilbert geometry with a mild condition. Also we show that the asymptotic dimension of any two-dimensional Hilbert geometry is just two. This implies that the coarse Baum-Connes conjecture holds for any two-dimensional Hilbert geometry via Yu's theorem.

math.MG↗

arXiv2014

Cannon-Thurston maps for Coxeter groups with signature $(n-1,1)$

For a Coxeter group $W$ we have an associating bi-linear form $B$ on suitable real vector space. We assume that $B$ has the signature $(n-1,1)$ and all the bi-linear form associating rank $n' (\ge 3)$ Coxeter subgroups generated by subsets of $S$ has the signature $(n',0)$ or $(n'-1,1)$. Under these assumptions, we see that there exists the Cannon-Thurston map for $W$, that is, the $W$-equivariant continuous surjection from the Gromov boundary of $W$ to the limit set of $W$. To see this we construct an isometric action of $W$ on an ellipsoid with the Hilbert metric. As a consequence, we see that the limit set of $W$ coincides with the set of accumulation points of roots of $W$.

math.GT↗

arXiv2014

Cannon-Thurston maps for Coxeter groups including affine special subgroups

For a Coxeter group $W$ we have an associating bi-linear form $B$ on a real vector space. We assume that $B$ has the signature $(n-1,1)$. In this case we have the Cannon-Thurston map for $W$, that is, a $W$-equivariant continuous surjection from the Gromov boundary of $W$ to the limit set of $W$. We focus on the case where Coxeter groups contain affine special subgroups.

math.GT↗