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Aleksander Skenderi

Publications and source records attributed to Aleksander Skenderi.

5 recordsLinked to original sources
arXiv2026

Growth gaps and generating sets

We show that the existence of a growth gap for infinite-index subgroups of a given finitely genrated group can depend on the finite generating set. More precisely, for any irreducible lattice $Λ$ in a higher rank semisimple Lie group $G$ with Kazhdan's property (T), the group $Λ\times Λ$ admits one finite symmetric generating set with a growth gap and another without a growth gap. We also prove that the growth gap can be made arbitrarily small. In contrast, for a non-elementary hyperbolic group the existence of a growth gap is independent of the finite generating set.

math.GR↗
arXiv2026

Zariski density of discrete subgroups via critical exponents and unitary representations

Let $G$ be a connected real semisimple linear algebraic group with finite center and no compact factors, let $X$ denote its associated Riemannian symmetric space, and let $h_{\mathrm{vol}}(X)$ be the volume growth entropy of $X$. We show that there exists an $ε= ε(G) > 0$ so that the following holds: if $Γ< G$ is a discrete subgroup with critical exponent greater than $h_{\mathrm{vol}}(X) - ε$, then $Γ$ is Zariski dense in $G$. If $G$ is further assumed to be isomorphic to one of the isometry groups of the real, complex, or quaternionic hyperbolic spaces, we determine the smallest possible value of the critical exponent to guarantee Zariski density of the associated subgroup (in other words, the largest possible value of $ε= ε(G)$). In the setting of discrete subgroups of real semisimple Lie groups with no compact factors, this generalizes Borel's Density Theorem for lattices. The key ingredient is input from the theory of unitary representations, particularly the recent work of Benoist--Liang (which was inspired by earlier work of Benoist--Kobayashi) on general temperedness criteria for the quasi-regular representation of $L^2(G/H)$ for any closed subgroup $H$ of $G$.

math.GR↗
arXiv2026

Free Semigroups of Large Critical Exponent

For a convergence group equipped with an expanding coarse-cocycle, we construct finitely generated free subsemigroups, which we call Bishop--Jones semigroups, of critical exponent arbitrarily close to but strictly less than the critical exponent of the ambient group. As an application, we show that for any non-elementary transverse subgroup $Γ$ of a semisimple Lie group $G$, there exist finitely generated free Anosov subsemigroups in the sense of Kassel--Potrie of critical exponent arbitrarily close to but strictly less than that of the ambient transverse group. Furthermore, we show that these semigroups admit quasi-isometric embeddings into the symmetric space X of G with certain additional coarse-geometric properties.

math.GR↗
arXiv2025

Asymptotically large free semigroups in Zariski dense discrete subgroups of Lie groups

Let $G$ be a connected algebraic semisimple real Lie group with finite center and no compact factors, and let $Γ$ be a Zariski dense discrete subgroup of $G$. We show that $Γ$ contains free, finitely generated subsemigroups whose critical exponents are arbitrarily close to that of $Γ$. Furthermore, these subsemigroups are Zariski dense in $G$ and $P$-Anosov in the sense of Kassel--Potrie. This shows that no gap phenomenon holds for critical exponents of discrete subsemigroups of Lie groups, which is in contrast with Leuzinger's critical exponent gap theorem for infinite covolume discrete subgroups of Lie groups with Kazhdan's property (T), proven in 2003. As an application, we prove that the critical exponent is lower semicontinuous in the Chabauty topology, in the following sense: if a sequence of Zariski dense discrete subgroups $\{Γ_{n}\}$ of $G$ converges in the Chabauty topology to a Zariski dense discrete subgroup $Γ$, then $\liminf_{n \to \infty} δ(Γ_{n}) \geq δ(Γ)$.

math.GR↗