arXiv subjects

Kiran Parkhe

Publications and source records attributed to Kiran Parkhe.

5 recordsLinked to original sources

arXiv2015

When actions of amenable groups can be lifted to the universal cover

In the first part of this paper, we let $G$ be a finitely-generated amenable group such that $G/[G, G]$ is torsion-free. We suppose that $G$ acts by homeomorphisms homotopic to the identity on a manifold $M$, and give conditions on $M$ which imply that such an action must lift to an action on the universal cover $\tilde{M}$. The circle, all 2-manifolds except the open annulus, and most compact 3-manifolds satisfy these conditions. The proof uses a dynamical tool called homological rotation vectors, and Thurston's Geometrization Theorem in the latter case. On manifolds not satisfying our conditions, such actions really may fail to lift. In the second part, we try to understand the dynamical possibilities in the simplest case: $G = \mathbb{Z}^2$, and $M = \mathbb{A}$ is the open annulus. We show that if a $\mathbb{Z}^2$ action homotopic to the identity on $\mathbb{A}$ fails to lift to a $\mathbb{Z}^2$ action on the plane, and if the action satisfies one additional condition (which may not be necessary), the action is essentially similar to the one generated by $\bar{f_0}(θ, y) = (θ+ y, y)$ and $\bar{g_0}(θ, y) = (θ, y + 1)$.

math.GT↗

arXiv2014

Smoothing nilpotent actions on 1-manifolds

Let $M$ be a connected 1-manifold, i.e., $M = \R \cong (0, 1), [0, 1), [0, 1]$, or $S^1$, and let $\Homeo_+(M)$ (resp. $\Diff_+^1(M)$) be the group of orientation-preserving homeomorphisms (resp. $C^1$ diffeomorphisms) of $M$. It is a classical result that if $N$ is a finitely-generated, torsion-free nilpotent group, then there exist 1-1 homomorphisms $ϕ\colon N \to \Homeo_+(M)$. Farb and Franks show that, in fact, there exists a 1-1 homomorphism $N \to \Diff_+^1(M)$. In this paper we obtain a stronger result: every action $ϕ\colon N \to \Homeo_+(M)$ is topologically conjugate to an action $\tildeϕ\colon N \to \Diff_+^1(M)$.

math.DS↗

arXiv2012

Smooth gluing of group actions and applications

Let $M_1$ and $M_2$ be two $n$-dimensional smooth manifolds with boundary. Suppose we glue $M_1$ and $M_2$ along some boundary components (which are, therefore, diffeomorphic). Call the result $N.$ If we have a group $G$ acting continuously on $M_1,$ and also acting continuously on $M_2,$ such that the actions are compatible on glued boundary components, then we get a continuous action of $G$ on $N$ that stitches the two actions together. However, even if the actions on $M_1$ and $M_2$ are smooth, the action on $N$ probably will not be smooth. We give a systematic way of smoothing out the glued $G$-action. This allows us to construct interesting new examples of smooth group actions on surfaces, and to extend a result of Franks and Handel on distortion elements in diffeomorphism groups of closed surfaces to the case of surfaces with boundary.

math.DS↗

arXiv2012

Distortion for diffeomorphisms of surfaces with boundary

If $G$ is a finitely generated group with generators $\{g_1,..., g_s\}$, we say an infinite-order element $f \in G$ is a distortion element of $G$ provided that $\displaystyle \liminf_{n \to \infty} \frac{|f^n|}{n} = 0$, where $|f^n|$ is the word length of $f^n$ with respect to the given generators. Let $S$ be a compact orientable surface, possibly with boundary, and let $\Diff(S)_0$ denote the identity component of the group of $C^1$ diffeomorphisms of $S$. Our main result is that if $S$ has genus at least two, and $f$ is a distortion element in some finitely generated subgroup of $\Diff(S)_0$, then $\supp(μ) \subseteq \Fix(f)$ for every $f$-invariant Borel probability measure $μ$. Under a small additional hypothesis the same holds in lower genus. For $μ$ a Borel probability measure on $S$, denote the group of $C^1$ diffeomorphisms that preserve $μ$ by $\Diff_μ(S)$. Our main result implies that a large class of higher-rank lattices admit no homomorphisms to $\Diff_μ(S)$ with infinite image. These results generalize those of Franks and Handel to surfaces with boundary.

math.DS↗

arXiv2011

One-parameter families of circle diffeomorphisms with strictly monotone rotation number

We show that if $f \colon S^1 \times S^1 \to S^1 \times S^1$ is $C^2$, with $f(x, t) = (f_t(x), t)$, and the rotation number of $f_t$ is equal to $t$ for all $t \in S^1$, then $f$ is topologically conjugate to the linear Dehn twist of the torus $(1&1 0&1)$. We prove a differentiability result where the assumption that the rotation number of $f_t$ is $t$ is weakened to say that the rotation number is strictly monotone in $t$.

math.DS↗