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Sagar B. Kalane

Publications and source records attributed to Sagar B. Kalane.

8 recordsLinked to original sources
arXiv2026

Strongly Doubly Reversible Pairs in Quaternionic Unitary Group of Signature $(n,1)$

Let $\mathrm{PSp}(n,1)$ denote the isometry group of quaternionic hyperbolic $n$--space $\h^n$. A pair $(g_1,g_2)\in\mathrm{PSp}(n,1)^2$ is \emph{strongly doubly reversible} if $(g_1,g_2)$ and $(g_1^{-1},g_2^{-1})$ are simultaneously conjugate by an involution. Equivalently, there exist involutions $i_1,i_2,i_3\in\mathrm{PSp}(n,1)$ such that $g_1=i_1i_2$ and $g_2=i_1i_3$. We prove that the set of strongly doubly reversible pairs has Haar measure zero in $\mathrm{PSp}(n,1)\times\mathrm{PSp}(n,1)$. The same conclusion holds for $\mathrm{PSp}(n)\times\mathrm{PSp}(n)$ and $\mathrm{SU}(n,1)\times\mathrm{SU}(n,1)$ for $n\ge2$, and for $\mathrm{SO}_0(n,1)\times\mathrm{SO}_0(n,1)$ and $\mathrm{SO}(n+1)\times\mathrm{SO}(n+1)$ for $n\ge4$. In the compact rank-one case, every pair in $\mathrm{PSp}(1)$ is strongly doubly reversible, which gives a short proof of the theorem of Basmajian and Maskit that every pair in $\mathrm{SO}(4)$ is strongly doubly reversible. For hyperbolic pairs, we prove that double reversibility and strong double reversibility are equivalent in $\mathrm{PSp}(1,1)$. In higher dimension, we prove the same implication when one member of the pair is regular, with pairwise distinct non-real unit eigenvalue classes. Finally, in $\mathrm{PSp}(1,1)$, for a hyperbolic element in normal form, we give a complete explicit matrix criterion characterizing all elements that form a strongly doubly reversible pair with it.

math.GR↗
arXiv2026

Trace Coordinates and Local Fenchel--Nielsen Parameters in Real Hyperbolic 5-Space

We study representations of fundamental groups of closed orientable surfaces into \(\mathrm{SL}(2,\mathbb{H})\), whose projectivization \(\mathrm{PSL}(2,\mathbb{H})\) is the group of orientation-preserving isometries of real hyperbolic \(5\)-space, with the aim of developing a quaternionic analogue of Fenchel--Nielsen theory. We give a normal form for pairs of regular loxodromic elements with disjoint fixed point sets and show that, on an open dense generic locus, the corresponding moduli space is \(15\)-dimensional. We also associate fifteen word traces whose differentials are linearly independent on an open dense subset and hence give local real-analytic coordinates there. For a pair of pants with prescribed regular loxodromic boundary conjugacy classes, we show that the relative deformation space is locally \(6\)-dimensional. Combining this internal pants data with the three-dimensional boundary conjugacy data and the three-dimensional centralizer gluing freedom gives a local parameter decomposition for closed surface group representations. For a closed surface of genus \(g\), this yields \(30g-30\) real parameters.

math.GT↗
arXiv2022

Free groups generated by two parabolic maps

In this paper we consider a group generated by two unipotent parabolic elements of ${\rm SU}(2,1)$ with distinct fixed points. We give several conditions that guarantee the group is discrete and free. We also give a result on the diameter of a finite ${\mathbb R}$-circle in the Heisenberg group.

math.GT↗
arXiv2021

On Free Group Generated by Two Heisenberg Translations

In this paper, we will discuss the groups generated by two Heisenberg translations of ${\rm PSp}(2,1)$ and determine when they are free. We improve a result given in \cite{xwy} by Xie, Wang, Jiang in Canad. Math. Bull. $56(2013), 881-889.$ and from that derive result for the ${\rm PSp}(2,1)$ case.

math.GR↗
arXiv2020

Local coordinates for complex and quaternionic hyperbolic pairs

Let $G(n)={\rm Sp}(n,1)$ or ${\rm SU}(n,1)$. We classify conjugation orbits of generic pairs of loxodromic elements in $G(n)$. Such pairs, called `non-singular', were introduced by Gongopadhyay and Parsad for ${\rm SU}(3,1)$. We extend this notion and classify $G(n)$-conjugation orbits of such elements in arbitrary dimension. We prove that the set given by non-singular pairs in $G(n)$ is `small' for $n \geq 4$. However, for $n=3$, they give a subspace that can be parametrized using a set of coordinates whose local dimension equals the dimension of the underlying group. We further construct twist-bend parameters to glue such representations and obtain local parametrization for generic representations of the fundamental group of a closed oriented surface into $G(3)$.

math.GT↗
arXiv2019

On Conjugation orbits of semisimple pairs in rank one

We consider Lie groups ${\rm SU}(n,1)$ and ${\rm Sp}(n,1)$ that act as the isometries of the complex and quaternionic hyperbolic spaces respectively. We classify pairs of semisimple elements in ${\rm Sp}(n,1)$ and ${\rm SU}(n,1)$ up to conjugacy. This gives local parametrization of the representations $ρ$ in $Hom(F_2, G)/G$ such that both $ρ(x)$ and $ρ(y)$ are hyperbolics, where $F_2=\langle x, y\rangle$, $G={\rm Sp}(n,1)$ or ${\rm SU}(n,1)$. We use the ${\rm PSp}(n,1)$-configuration space $M(n,i,m-i)$ of ordered $m$-tuples of points on $\overline{{\bf H}_{\mathbb H}^n}$, where first $i$ points in an $m$-tuple are null points, to classify the semisimple pairs. Further, we also classify points on $M(n,i,m-i)$.

math.GT↗
arXiv2018

Quaternionic Hyperbolic Fenchel-Nielsen Coordinates

Let $Sp(2,1)$ be the isometry group of the quaternionic hyperbolic plane ${{\bf H}_{\mathbb H}}^2$. An element $g$ in $Sp(2,1)$ is `hyperbolic' if it fixes exactly two points on the boundary of ${{\bf H}_{\mathbb H}}^2$. We classify pairs of hyperbolic elements in $Sp(2,1)$ up to conjugation. A hyperbolic element of $Sp(2,1)$ is called `loxodromic' if it has no real eigenvalue. We show that the set of $Sp(2,1)$ conjugation orbits of irreducible loxodromic pairs is a $(\mathbb C {\mathbb P}^1)^4$-bundle over a topological space that is locally a semi-analytic subspace of ${\mathbb R}^{13}$. We use the above classification to show that conjugation orbits of `geometric' representations of a closed surface group (of genus $g \geq 2$) into $Sp(2,1)$ can be determined by a system of $42g-42$ real parameters. Further, we consider the groups $Sp(1,1)$ and $GL(2, {\mathbb H})$. These groups also act by the orientation-preserving isometries of the four and five dimensional real hyperbolic spaces respectively. We classify conjugation orbits of pairs of hyperbolic elements in these groups. These classifications determine conjugation orbits of `geometric' surface group representations into these groups.

math.GT↗