arXiv subjects

Soumyadip Thandar

Publications and source records attributed to Soumyadip Thandar.

5 recordsLinked to original sources

arXiv2026

Linear-growth harmonic functions for nonsymmetric random walks on groups of polynomial growth

Let $G$ be a finitely generated group of polynomial growth and let $μ$ be an adapted, Abelian-centered probability measure with a finite exponential moment (not necessarily symmetric or finitely supported). We prove that every $μ$-harmonic function of at most linear growth is globally Lipschitz. In particular, if $G$ is nilpotent, these functions are exactly the affine characters: $$ \mathrm{HF}_1(G,μ)=\operatorname{LHF}(G,μ)=P^1(G). $$ The analytic machinery required for this is a convolution-gradient estimate that is uniform over families of Abelian-centered, finitely supported probability measures satisfying fixed ellipticity and exponential-moment bounds. A quantitative induction-restriction theorem transfers $\operatorname{LHF}$ across finite-index subgroups. We also determine the normed structure of these spaces. In particular, on a nilpotent group, the Lipschitz seminorm of an affine character is exactly the dual stable norm of the word metric. Under the same nonsymmetric hypotheses, linear-growth harmonic functions modulo constants identify canonically with the virtual first cohomology. This identification is an isometry for the asymptotic Lipschitz seminorm. For the ordinary Lipschitz seminorm it is a contraction with bounded inverse; the inverse bound seems to depend on $μ$, as shown by an infinite dihedral group example.

math.GR↗

arXiv2026

Higher topological complexity of Seifert fibered manifolds

In this article, we investigate the higher topological complexity of oriented Seifert fibered manifolds that are Eilenberg--MacLane spaces $K(G,1)$ with infinite fundamental group $G$. We first refine the cohomological lower bounds for higher topological complexity by introducing the notion of higher topological complexity weights. As an application, we show that the $r^{\text{th}}$ topological complexity of these manifolds lies in $\{3r-1, 3r, 3r+1\}$, and characterize large families where the value is $3r$ or $3r+1$. Additionally, we establish a sufficient condition for higher topological complexity to be exactly $3r$ when the base surface is orientable and aspherical. Finally, we show that the higher topological complexity of the wedge of finitely many closed, orientable, aspherical $3$-manifolds is exactly $3r+1$.

math.AT↗

arXiv2026

Green geometry, Martin boundary and random walk asymptotics on groups

We identify a single computationally checkable analytic quantity interlacing Martin boundary collapse, Green geometry, and linear escape for transient random walks on finitely generated groups: the Green-variation functional \[ Δ(S;a,b):=\max_{x\in\partial S}\frac{|G(a,x)-G(b,x)|}{G(a,x)}. \] We prove that $Δ\to0$ along exhaustions characterises the strong Liouville property (under mild, verifiable hypotheses on the ``strong Liouville $\Rightarrow Δ\to0$'' direction), turning boundary oscillation estimates for Green kernels into potential-theoretic rigidity. We then give two general criteria for $Δ$-vanishing. The first one derives quantitative bounds on $Δ$ from coarse heat-kernel envelopes at an intrinsic scale together with a Tauberian comparability, covering Gaussian/sub-Gaussian and stable-like regimes; and the second one is purely elliptic: an ``elliptic Hölder exhaustion'' criterion. Conversely, on groups of exponential growth, $Δ$ fails to decay along balls already under stretched-exponential on-diagonal upper bounds, yielding a quantitative obstruction to strong Liouville. As consequences, trivial Martin boundary forces linear-scale collapse of Green geometry ($d_G(e,x)=o(|x|)$) and vanishing Green speed (in probability), without any entropy hypothesis. On the non-Liouville side we prove an abundance principle: the existence of a single minimal positive harmonic function at a prescribed growth scale forces infinitely many. Finally, we clarify the role of moment assumptions in speed theory: any linear-speed law of large numbers on a set of positive probability forces $\mathbb E|X_1|<\infty$, while on torsion-free nilpotent groups one can have $\mathbb E|X_1|=\infty$ yet $|X_n|/n\to0$ in probability.

math.GR↗

arXiv2025

Equivariant Intrinsic Formality

Algebraic models for equivariant rational homotopy theory were developed by Triantafillou and Scull for finite group actions and $S^1$ action, respectively. They showed that given a diagram of rational cohomology algebras from the orbit category of a group $G$, there is a unique minimal system of DGAs and hence a unique equivariant rational homotopy type that is weakly equivalent to it. However, there can be several equivariant rational homotopy types with the same system of cohomology algebras. Halperin, Stasheff, and others studied the problem of classifying rational homotopy types up to cohomology in the non-equivariant case. In this article, we consider this question in the equivariant case. We prove that when $\mathbb{Z}_p$ under suitable conditions, the equivariant rational homotopy types with isomorphic cohomology can be reduced to the non-equivariant case.

math.AT↗

arXiv2025

$C_{pq}$-Injective Diagrams and a Combination Theorem for Minimal Models

We study diagrams of commutative differential graded algebras (DGAs) over the orbit category $\sO_G$ in the context of equivariant rational homotopy theory. For $G = C_{pq}$ with $p, q$ distinct primes, we give necessary conditions for injectivity. We prove a combination-type result: the equivariant wedge of injective diagrams over $\mathcal{O}_{C_p}$ and $\mathcal{O}_{C_q}$ with retract structure maps yields an injective diagram over $\mathcal{O}_{C_{pq}}$ with a level-wise minimal model. As an application, we construct examples of $C_{pq}$-formal spaces.

math.AT↗