Searcharxiv⌕ Search

arXiv subjects

Konstantinos Tsouvalas

Publications and source records attributed to Konstantinos Tsouvalas.

At least 19 recordsLinked to original sources
arXiv2026

On topologically transitive subsets of flag manifolds

We discuss some contexts in which the topological dynamics of certain Anosov subgroups of higher-rank Lie groups on "bad" subsets of flag manifolds can be analyzed using results from homogeneous dynamics in the infinite-covolume rank-one setting. For example, we show that a group of projective transformations dividing a strictly convex domain in projective space and intersecting Zariski-densely the stabilizer of an ellipsoid has a dense orbit in the complement of the domain, and acts minimally on the space of full projective flags tangent to the domain. This accounts for all known examples of divisible strictly convex domains in sufficiently high dimensions. We also provide examples of Zariski-dense groups that are Anosov in a partial flag manifold but such that the equivariant projection from the Benoist-Guivarc'h limit set in the Furstenberg boundary to the Anosov limit set is not a fibration.

math.GT↗
arXiv2026

Robust quasi-isometric embeddings inapproximable by Anosov representations

Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. For all but finitely many $m\in \mathbb{N}$, we exhibit the first examples of non-locally rigid, Zariski dense, robust quasi-isometric embeddings of hyperbolic groups in $\mathsf{SL}_m(\mathbb{K})$ which are not limits of Anosov representations. As a consequence, we show that higher rank analogues of Sullivan's structural stabilty theorem and of the density theorem for Kleinian groups fail for Anosov representations in $\mathsf{SL}_m(\mathbb{C}), m\geq 30$.

math.GR↗
arXiv2026

Ping-pong in the projective plane over a nonarchimedean field

We show that any lattice in $\mathrm{SL}_3(k)$, where $k$ is a nonarchimedean local field, contains an undistorted subgroup isomorphic to the free product $\mathbb{Z}^2*\mathbb{Z}$. To our knowledge, the subgroups we construct give the first examples in the literature of finitely generated discrete subgroups of nonarchimedean Lie groups that are not virtually isomorphic to lattices in such Lie groups. Our result is in contrast to the case of $\mathrm{SL}_3(\mathbb{Z})$, in which the existence of a $\mathbb{Z}^2*\mathbb{Z}$ subgroup remains open.

math.GR↗
arXiv2026

Matrix entries, unipotents, and linearity of amalgams

We investigate linearity of amalgams of subgroups of algebraic groups along intersections with algebraic subgroups. In the process, we establish linearity of certain "doubles" of linear groups, and obtain new examples of finitely generated residually finite groups that fail to be linear.

math.GR↗
arXiv2026

On transversality in flag manifolds and linearity of amalgams

We show that the fundamental group of the double of a complete negatively curved locally symmetric manifold along a closed geodesic is linear. More generally, we establish linearity of doubles of torsion-free transverse subgroups (also known in the literature as regular antipodal subgroups) of semisimple Lie groups along biproximal maximal cyclic subgroups.

math.GR↗
arXiv2026

Directional growth of coamenable normal subgroups: counterexamples and rigidity

Roblin's theorem asserts that, in rank one, a coamenable normal subgroup has the same critical exponent as its ambient group. Natural higher-rank analogues would predict that coamenability preserves the limit cone and the growth indicator. We show that both assertions fail, even when the quotient is infinite cyclic. For every odd integer $n\geq 3$, we construct a nonempty open family of Zariski-dense Borel--Anosov Schottky subgroups of $\mathrm{SL}_n(\mathbb R)$ admitting cocyclic normal subgroups with strictly smaller limit cones. Moreover, in $\mathrm{SL}_3(\mathbb R)$, we construct cocyclic pairs with the same limit cone but distinct growth indicators at an interior direction. We then identify the precise rigidity that survives. Let $Γ$ be a Zariski-dense Borel--Anosov subgroup of a connected semisimple real algebraic group, and let $N\lhdΓ$ be coamenable. Then the growth indicators of $N$ and $Γ$ agree on the fixed-point set of the opposition involution, and their Riemannian critical exponents are equal. The examples with equal limit cones show that the restriction to opposition-invariant directions is sharp.

math.GR↗
arXiv2026

Robust quasi-isometric embeddings of virtually free groups

Let $k$ be a nonarchimedean local field. For any $n\geq 3$, we construct the first examples of robust quasi-isometric embeddings of non-elementary free groups into $\mathsf{GL}_n(k)$ which are not limits of Anosov representations. If $\bf{K}=\mathbb{R},\mathbb{C}$, we exhibit examples of non-locally rigid, robust quasi-isometric embeddings of virtually free groups into $\mathsf{GL}_n(\bf{K})$, $n\geq 3$, which are not limits of Anosov representations. Moreover, we exhibit a non-Anosov robust quasi-isometric embedding of the free semigroup $\mathbb{Z}\ast \mathbb{Z}^{+}$ into $\mathsf{GL}_3(\mathbb{C})$, which is a limit of Anosov representations.

math.GR↗
arXiv2025

Undecidability of epimorphisms onto products of hyperbolic groups

We exhibit examples of finitely presented subgroups $P$ of direct products of hyperbolic groups for which there is no algorithm that detects whether a finitely presented group has a quotient isomorphic to $P$. For any torsion-free, linear, hyperbolic group $Q$ that maps onto the free group of rank $2$ and $m\geq 2$, we construct a recursive sequence $(Γ_n)_{n\in \mathbb{N}}$ of torsion-free, hyperbolic $C'(\frac{1}{6})$ small cancellation groups, with the property that there is no algorithm determining the values $n\in \mathbb{N}$ such that $Γ_n$ has a quotient isomorphic to the direct product $Q^{m}$ of $m$-copies of $Q$.

math.GR↗
arXiv2025

Anosov representations, strongly convex cocompact groups and weak eigenvalue gaps

We provide characterizations of Anosov representations of word hyperbolic groups into real semisimple Lie groups in terms of the existence of equivariant limit maps on the Gromov boundary, the Cartan property and the uniform gap summation property introduced by Guichard-Guéritaud-Kassel-Wienhard. We also study representations of finitely generated groups satisfying weak uniform gaps in eigenvalues and establish conditions to be Anosov. As an application, we also obtain a characterization of strongly convex cocompact subgroups of the projective linear group $\mathsf{PGL}_d(\mathbb{R})$.

math.GT↗
arXiv2025

The Hölder exponent of Anosov limit maps

Let $Γ$ be a non-elementary word hyperbolic group and $d_{a}, a>1,$ a visual metric on its Gromov boundary $\partial_{\infty}Γ$. For an $1$-Anosov representation $ρ:Γ\rightarrow \mathsf{GL}_{d}(\mathbb{K})$, where $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, we calculate the Hölder exponent of the Anosov limit map $ξ_ρ^1:(\partial_{\infty}Γ, d_{a})\rightarrow (\mathbb{P}(\mathbb{K}^d),d_{\mathbb{P}})$ of $ρ$ in terms of the moduli of eigenvalues of elements in $ρ(Γ)$ and the stable translation length on $Γ$. If $ρ$ is either irreducible or $ξ_ρ^1(\partial_{\infty}Γ)$ spans $\mathbb{K}^d$ and $ρ$ is $\{1,2\}$-Anosov, then $ξ_ρ^1$ attains its Hölder exponent. We also provide an analogous calculation for the exponent of the inverse limit map of $(1,1,2)$-hyperconvex representations. Finally, we exhibit examples of non semisimple $1$-Anosov representations of surface groups in $\mathsf{SL}_4(\mathbb{R})$ whose Anosov limit map in $\mathbb{P}(\mathbb{R}^4)$ does not attain its Hölder exponent.

math.DS↗
arXiv2025

Anosov representations of amalgams

For uniform lattices $Γ$ in rank 1 Lie groups, we construct Anosov representations of virtual doubles of $Γ$ along certain quasiconvex subgroups. We also show that virtual HNN extensions of these lattices over some cyclic subgroups admit Anosov embeddings. In addition, we prove that for any Anosov subgroup $Γ$ of a real semisimple linear Lie group $\mathsf{G}$ and any infinite abelian subgroup $\mathrm{H} $ of $ Γ$, there exists a finite-index subgroup $Γ' $ of $ Γ$ containing $\mathrm{H}$ such that the double $Γ' *_{\mathrm{H}} Γ'$ admits an Anosov representation, thereby confirming a conjecture of [arXiv:2112.05574]. These results yield numerous examples of one-ended hyperbolic groups that do not admit discrete and faithful representations into rank 1 Lie groups but do admit Anosov embeddings into higher-rank Lie groups.

math.GR↗
arXiv2024

Subgroups of CAT(0) groups, exotic finiteness properties and non-QI-embeddings into linear groups

For every positive integer $n$ we construct an example of a subgroup $L< G$ of a linear ${\rm CAT}(0)$ group $G$ such that $L$ is of finiteness type $\mathcal{F}_{n-1}$ and not $\mathcal{F}_n$, and $L$ does not admit a representation into $\mathsf{GL}_d(k)$ which is a quasi-isometric embedding for any local field $k$. We further prove that there is a faithful representation of $L$ into some $\mathsf{GL}_{\ell}(\mathbb{C})$ which is not the restriction of any representation of $G$. This generalises a family of fibre products of type $\mathcal{F}_2$ not $\mathcal{F}_3$ with these properties constructed by the second author.

math.GR↗
arXiv2024

Singular value gap estimates for free products of semigroups

We establish lower estimates for singular value gaps of free products of $1$-divergent semigroups $Γ_1,Γ_2\subset \mathsf{GL}_d(\mathbb{K})$ which are in ping-pong position. As an application, we prove that if $Γ_1$ and $Γ_2$ are quasi-isometrically embedded subgroups in ping pong position, then the group they generate $\langle Γ_1,Γ_2\rangle$ is also quasi-isometrically embedded. In addition, we establish that the class of linear finitely generated groups, admitting a faithful linear representation over $\mathbb{R}$ which is a quasi-isometric embedding, is closed under free products.

math.GR↗
arXiv2024

Topological restrictions on relatively Anosov representations

We obtain restrictions on which groups can admit relatively Anosov representations into specified target Lie groups, by examining the topology of possible Bowditch boundaries and how they interact with the Anosov limit maps. For instance, we prove that, up to finite index, any group admitting a relatively Anosov representation into SL(3,R) is a free group or surface group, and any group admitting a relatively k-Anosov representation into Sp(2m,R), where k is an odd integer between 1 and m, is a surface group or a free product of nilpotent groups. We also obtain a characterization of groups admitting relatively 1-Anosov representations into SL(4,R), general bounds on the dimension of the Bowditch boundary of groups admitting relatively Anosov representations into SL(d,R), statements relating spheres in the Bowditch boundary to the (non-)existence of relatively Anosov representations, and a characterization of groups of cohomological dimension at least d-1 admitting relatively 1-Anosov representations into SL(d,R).

math.GR↗
arXiv2024

Cartan projections of fiber products and non quasi-isometric embeddings

Let $Γ$ be a finitely generated group and $N$ be a normal subgroup of $Γ$. The fiber product of $Γ$ with respect to $N$ is the subgroup $Γ\times_N Γ=\{(γ, γw): γ\in Γ, w \in N\}$ of the direct product $Γ\times Γ$. For every representation $ρ:Γ\times_N Γ\rightarrow \mathsf{GL}_d(k)$, where $k$ is a local field, we establish upper bounds for the norm of the Cartan projection of $ρ$ in terms of a fixed word length function on $Γ$. As an application, we exhibit examples of finitely generated and finitely presented fiber products $P=Γ\times_N Γ$, where $Γ$ is linear and Gromov hyperbolic, such that $P$ does not admit linear representations which are quasi-isometric embeddings.

math.GR↗
arXiv2023

Anosov groups that are indiscrete in rank one

We exhibit Anosov subgroups of $\mathsf{SL}_d(\mathbb{R})$ that do not embed discretely in any rank-$1$ simple Lie group of noncompact type, or indeed, in any finite product of such Lie groups. These subgroups are isomorphic to free products $Γ* Δ$, where $Γ$ is a uniform lattice in $\mathsf{F}_4^{(-20)}$ and $Δ$ is a uniform lattice in $\mathsf{Sp}(m,1)$, $m \geq 51$.

math.GR↗
arXiv2023

On regular subgroups of $\mathsf{SL}_3(\mathbb{R})$

Motivated by a question of M. Kapovich, we show that the $\mathbb{Z}^2$ subgroups of $\mathsf{SL}_3(\mathbb{R})$ that are regular in the language of Kapovich--Leeb--Porti, or divergent in the sense of Guichard--Wienhard, are precisely the lattices in minimal horospherical subgroups. This rules out any relative Anosov subgroups of $\mathsf{SL}_3(\mathbb{R})$ that are not in fact Gromov-hyperbolic. By work of Oh, it also follows that a Zariski-dense discrete subgroup $Γ$ of $\mathsf{SL}_3(\mathbb{R})$ contains a regular $\mathbb{Z}^2$ if and only if $Γ$ is commensurable to a conjugate of $\mathsf{SL}_3(\mathbb{Z})$. In particular, a Zariski-dense regular subgroup of $\mathsf{SL}_3(\mathbb{R})$ contains no $\mathbb{Z}^2$ subgroups.

math.GR↗