Bordered hyperbolic manifolds with a fixed perimeter-to-volume ratio

Franco Vargas Pallete; Sami Douba

Bordered hyperbolic manifolds with a fixed perimeter-to-volume ratio

T0 review · 0 major / 5 minor · reviewed 2026-07-11 · grok-4.5

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Infinitely many incommensurable hyperbolic n-manifolds can share one perimeter-to-volume ratio.

desk verdict Clean existence note: infinite incommensurable families of bordered hyperbolic n-manifolds (n≥3) sharing one perimeter-to-volume ratio, via standard GPS hybrids. read the letter →

arxiv 2607.05232 v1 pith:VEKMR7VV submitted 2026-07-06 math.GT

Sami Douba , Franco Vargas Pallete This is my paper · ORCID

classification math.GT MSC 57M5022E4053C22

keywords hyperbolicmanifoldstotallygeodesicboundaryperimeter-to-volumeratiocommensurabilityarithmetichybridsSchottkysets

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Among finite-volume hyperbolic manifolds of dimension three and higher that carry a nonempty totally geodesic boundary, the ratio of boundary area to volume is a natural commensurability invariant. This paper shows that the invariant is coarse: for every dimension n at least 3 one can build infinite families of pairwise incommensurable examples, both compact and cusped, that all realize exactly the same ratio. The manifolds are hybrids obtained by cutting arithmetic hyperbolic manifolds along carefully chosen totally geodesic hypersurfaces and then regluing the pieces according to different patterns. A comparison lemma then guarantees that distinct gluing patterns produce distinct commensurability classes, even though the perimeter-to-volume ratio stays fixed. The construction therefore demonstrates that this classical geometric ratio alone cannot separate commensurability classes of bordered hyperbolic manifolds.

What carries the argument

Arithmetic hybrids in the style of Gromov–Piatetski-Shapiro, cut and reglued along totally geodesic hypersurfaces, together with a comparison lemma that recovers the commensurability class of the arithmetic building blocks from any common finite cover of the hybrids.

What would settle it

Exhibit a common finite cover of two of the constructed hybrids N_j and N_k (j different from k) whose arithmetic doubles remain incommensurable, or compute that their perimeter-to-volume ratios actually differ.

Watch

Extended reading notes

Core claim

For every n greater than or equal to 3 there exist infinitely many pairwise incommensurable compact (respectively, noncompact complete finite-volume) hyperbolic n-manifolds with nonempty totally geodesic boundary that all share one and the same perimeter-to-volume ratio.

Load-bearing premise

The argument needs that a Zariski-dense subgroup of an arithmetic lattice already determines the lattice’s commensurability class, and for dimension three that the relevant boundary pieces stay compact so density still holds.

Editorial extensions

If this is right

Perimeter-to-volume ratio alone cannot classify commensurability classes of bordered hyperbolic n-manifolds for n greater than or equal to 3.
The same fixed ratio can be realized by both compact and cusped infinite families.
The common ratio can be made arbitrarily small by passing to deeper congruence covers.
The associated Schottky sets of the constructed families are pairwise non-quasisymmetric.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Hausdorff dimension of the limit set may still separate the same families even when the ratio does not.
Similar hybrid constructions could produce infinite families sharing other coarse geometric invariants such as volume entropy or bottom of the spectrum.
The method suggests that many other arithmetic-sensitive invariants of bordered manifolds are likewise coarse.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This is a short, explicit construction that settles a natural question: the perimeter-to-volume ratio is a coarse commensurability invariant even among finite-volume hyperbolic n-manifolds with totally geodesic boundary. For every n≥3 they produce infinite pairwise incommensurable families, both compact and cusped, all with the same ratio.

What is new is the systematic production of those families. The ingredients are familiar: Gromov–Piatetski-Shapiro hybrids of arithmetic pieces, non-similar quadratic forms taken from Gelander–Levit, Lubotzky-type congruence covers that give non-separating hypersurfaces, and Raimbault’s comparison lemma that distinguishes the hybrids by looking at the pure sub-pieces. They compute the common ratio directly from the volumes and areas of the blocks, then vary the number of copies so that the pure N^{1}_{j} pieces have strictly decreasing ratios; Lemma 2 then forces the hybrids to be pairwise incommensurable. The argument is elementary once the classical arithmetic and Zariski-density facts are granted, and they flag the n=3 compactness hypothesis needed for density of pieces with corners.

Soft spots are minor and classical. Existence of a suitable congruence level m is standard; the density claim in Lemma 2 is textbook for finite-volume hyperbolic manifolds (and for n=3 with compact boundary). No circularity, no free parameters that affect the conclusion, and the citation pattern is honest. The result is solid but of limited scope—it does not produce maximizers or control Hausdorff dimension of the Schottky sets, both of which the authors themselves note remain open.

This is for people who work on arithmetic hyperbolic geometry, commensurability invariants, or Schottky sets. A serious referee will verify the density statements and the non-similarity of the forms, but the paper is already short and carefully written. I would accept it for peer review without hesitation and would cite the existence statement if I needed an example of how coarse this particular invariant is.

Referee Report

0 major / 5 minor

Summary. The paper proves that for every n≥3 there exist infinitely many pairwise incommensurable compact (resp. noncompact complete finite-volume) hyperbolic n-manifolds with nonempty totally geodesic boundary that all share one and the same perimeter-to-volume ratio (Theorem 1). The construction proceeds by cutting arithmetic congruence manifolds associated to carefully chosen quadratic forms f1,f2 (anisotropic over Q(√2) for the compact case; isotropic over Q for the cusped cases, with a mixed pair when n=3) along pairs of isometric totally geodesic hypersurfaces, then hybridizing the resulting pieces N^{1}_{j} and N^{2}_{j} in the style of Gromov–Piatetski-Shapiro so that the hybrids Nj have identical area(∂Nj)/vol(Nj). Incommensurability of the Nj is deduced from a comparison lemma (Lemma 2, after Raimbault) that recovers the pure arithmetic pieces from any common finite cover, using that Zariski-dense subgroups already determine the commensurability class of an arithmetic lattice (with an extra compactness hypothesis on the boundary when n=3).

Significance. The perimeter-to-volume ratio is an elementary commensurability invariant of finite-volume hyperbolic manifolds with totally geodesic boundary. The paper shows that this invariant is nevertheless crude: a single value is realized by infinitely many distinct commensurability classes in every dimension n≥3, both compact and cusped. The argument is short, explicit, and relies only on standard tools (Borel–Harish-Chandra, Mostow–Prasad, Lubotzky congruence subgroups, non-similarity of quadratic forms as in Gelander–Levit, and classical Zariski-density for finite-volume hyperbolic manifolds with corners). The construction therefore supplies a clean negative answer to the natural question of whether the ratio separates commensurability classes, and it places the result in the broader context of Schottky-set rigidity and random hyperbolic 3-manifolds.

minor comments (5)

In the proof of Theorem 1 the ideal m is asserted to exist by “an argument of Lubotzky [14] (see also [8])” so that two lifts of Σ i(n) fail to separate Mi(m). A one-sentence reminder of the free-quotient / first Betti-number step would make the existence self-contained for readers who do not immediately recall the reference.
The footnote on Zariski density for n=3 with compact boundary is essential to Lemma 2; it would be clearer if the same statement were also recorded in the body of the lemma rather than only in a footnote.
Figure 1 is a useful schematic of the hybrid gluing, but the caption does not explain the colouring (red/green boundary components) used in the surrounding text; a short parenthetical would help.
The final remarks mention that the common ratio can be made arbitrarily small by diminishing m (citing Belolipetsky–Weinberger). It would be useful to note explicitly that the same argument yields ratios arbitrarily close to the Miyamoto maximizers when one starts from the Apollonian lattice, even if that is left for future work.
Typographical: the title of the arXiv abstract has a line-break hyphen in “RA TIO”; the published version should of course remove it. Also, “O′fi(R)” is introduced without a prior definition of the full orthogonal group Ofi(R).

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: explicit hybrid construction with externally justified arithmetic incommensurability and elementary ratio computation.

full rationale

The paper constructs the manifolds Nj by gluing finitely many copies of two arithmetic pieces Mi cut along geodesic hypersurfaces, so that area(∂Nj)=2^{j+1} area(Σ1) and vol(Nj)=2^{j-1}(vol(M1)+vol(M2)) by direct counting; the common ratio is therefore immediate from the volumes and areas of the fixed building blocks and is not defined in terms of any target. Incommensurability of the Nj follows from Lemma 2 (sketched from Raimbault) applied to the pure subpieces N^{1}_j whose own ratios decrease with j, together with the non-similarity of the quadratic forms f1,f2 over K (Gelander–Levit) that makes the doubles 2N^{1}_j and 2N^{2}_j incommensurable arithmetic lattices. All load-bearing arithmetic, density, and rigidity ingredients are classical or external; the single self-citation [8] is only an alternative reference for a Lubotzky-type congruence argument and is not required for the central claim. No quantity is fitted, no uniqueness theorem is imported from the authors, and no prediction reduces to an input by construction.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The paper is a pure-existence construction inside arithmetic hyperbolic geometry. It relies on standard theorems about arithmetic lattices, congruence subgroups and Zariski density; the only free choices are the concrete quadratic forms (taken from Gelander–Levit) and the congruence levels whose existence is guaranteed by Lubotzky. No new physical or geometric entities are postulated.

free parameters (2)

choice of quadratic forms f1,f2
The concrete coefficients (17, 5, 7, etc.) are selected so that the forms are not similar over the base field; any other pair of non-similar forms with matching lower coefficients would serve equally well.
congruence ideal m
A sufficiently deep congruence level is chosen so that the hypersurfaces become two-sided, embedded and non-separating; existence is guaranteed by Lubotzky but the precise ideal is not computed.

assumptions (5)

standard math Borel–Harish-Chandra theorem: arithmetic quotients of anisotropic forms are compact.
Used to guarantee compactness of the Mi when the forms are K-anisotropic.
standard math Mostow–Prasad rigidity: complete finite-volume hyperbolic n-manifolds (n≥3) are determined up to isometry by their fundamental groups.
Implicit background that makes commensurability a well-defined geometric relation.
domain assumption A Zariski-dense subgroup of an arithmetic lattice in Isom(H^n) determines the commensurability class of the lattice.
Load-bearing for Lemma 2; standard but not proved in the paper.
domain assumption Lubotzky’s congruence-subgroup argument produces non-separating pairs of lifts of a totally geodesic hypersurface.
Cited to obtain the manifolds Mi with four boundary components.
domain assumption The chosen pairs of quadratic forms are not similar over the base field K.
Taken from Gelander–Levit (or verified by anisotropy for n=3); guarantees that the doubles 2N1j and 2N2j are incommensurable.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Bordered hyperbolic manifolds with a fixed perimeter-to-volume ratio." pith.science (2026). https://pith.science/paper/VEKMR7VV

@misc{pith2026260705232,
  author       = {Pith},
  title        = {Pith review of: Bordered hyperbolic manifolds with a fixed perimeter-to-volume ratio},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VEKMR7VV}},
  note         = {Machine review of arXiv:2607.05232}
}

abstract

We exhibit, for any $n \geq 3$, infinitely many pairwise incommensurable complete finite-volume hyperbolic $n$-manifolds, both compact and noncompact, each with nonempty totally geodesic boundary and all sharing the same perimeter-to-volume ratio.

Figures

Figures reproduced from arXiv: 2607.05232 by the authors.

**Figure 1.** Schematic representation of N1 and N2, respectively. with entirely red boundary obtained by gluing the Ni j along the green boundary components via isometries (see [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

300 extracted references · 111 canonical work pages

[1]

Journal of Topology , volume=

Virtual cubulation of nonpositively curved graph manifolds , author=. Journal of Topology , volume=. 2013 , publisher=

2013
[2]

Ergodic Theory Dynam

Rees, Mary , TITLE =. Ergodic Theory Dynam. Systems , FJOURNAL =. 1981 , NUMBER =. doi:10.1017/s0143385700001206 , URL =

work page doi:10.1017/s0143385700001206 1981
[3]

Geometry and dynamics of groups and spaces , SERIES =

Kapovich, Michael , TITLE =. Geometry and dynamics of groups and spaces , SERIES =. 2008 , MRCLASS =. doi:10.1007/978-3-7643-8608-5\_13 , URL =

doi:10.1007/978-3-7643-8608-5 2008
[4]

Discrete subgroups with finite Bowen-Margulis-Sullivan measure in higher rank

Fraczyk, Mikolaj and Lee, Minju , journal=. Discrete subgroups with finite Bowen-Margulis-Sullivan measure in higher rank
[5]

Aleksandrov, A. D. , TITLE =. Trudy Mat. Inst. Steklov. , VOLUME =. 1951 , PAGES =

1951
[6]

Semisimple

de Cornulier, Yves , journal=. Semisimple
[7]

Non-linear residually finite groups , JOURNAL =

Dru. Non-linear residually finite groups , JOURNAL =. 2005 , NUMBER =. doi:10.1016/j.jalgebra.2004.06.025 , URL =

work page doi:10.1016/j.jalgebra.2004.06.025 2005
[8]

Convex cocompactness for

Danciger, Jeffrey and Gu. Convex cocompactness for. arXiv preprint arXiv:2102.02757 , year=

arXiv

Show all 300 references

[9]

Breuillard, Emmanuel , TITLE =. Groups. 2015 , ISBN =

2015
[10]

Salehi and Varj\'u, P\'eter P

Golsefidy, A. Salehi and Varj\'u, P\'eter P. , TITLE =. Geom. Funct. Anal. , FJOURNAL =. 2012 , NUMBER =. doi:10.1007/s00039-012-0190-7 , URL =

2012 doi
[11]

arXiv preprint arXiv:2407.09439 , year=

Combination theorems in convex projective geometry , author=. arXiv preprint arXiv:2407.09439 , year=

arXiv
[12]

Convex cocompactness in pseudo-

Danciger, Jeffrey and Gu. Convex cocompactness in pseudo-. Geom. Dedicata , FJOURNAL =. 2018 , PAGES =. doi:10.1007/s10711-017-0294-1 , URL =

2018 doi
[13]

Wehrfritz, Bertram A F , TITLE =. Proc. London Math. Soc. (3) , FJOURNAL =. 1973 , PAGES =. doi:10.1112/plms/s3-27.3.402 , URL =

1973 doi
[14]

Wise, D. T. , TITLE =. Proceedings of the. 2014 , MRCLASS =

2014
[15]

Klingler, Bruno , TITLE =. Invent. Math. , FJOURNAL =. 2003 , NUMBER =. doi:10.1007/s00222-002-0283-2 , URL =

2003 doi
[16]

arXiv preprint arXiv:2006.03008 , year=

Arithmeticity, superrigidity and totally geodesic submanifolds of complex hyperbolic manifolds , author=. arXiv preprint arXiv:2006.03008 , year=

2006 arXiv
[17]

Indiana Univ

Stover, Matthew , TITLE =. Indiana Univ. Math. J. , FJOURNAL =. 2021 , NUMBER =. doi:10.1512/iumj.2021.70.8191 , URL =

2021 doi
[18]

Raimbault, Jean , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2013 , NUMBER =. doi:10.1093/imrn/rns151 , URL =

2013 doi
[19]

Gelander, Tsachik and Levit, Arie , TITLE =. Geom. Funct. Anal. , FJOURNAL =. 2014 , NUMBER =. doi:10.1007/s00039-014-0294-3 , URL =

2014 doi
[20]

Transform

Lubotzky, Alexander , TITLE =. Transform. Groups , FJOURNAL =. 1996 , NUMBER =. doi:10.1007/BF02587736 , URL =

1996 doi
[21]

Belolipetsky, Mikhail and Weinberger, Shmuel , TITLE =. Geom. Funct. Anal. , FJOURNAL =. 2024 , NUMBER =. doi:10.1007/s00039-024-00686-7 , URL =

2024 doi
[22]

and Thurston, W

Gromov, M. and Thurston, W. , TITLE =. Invent. Math. , FJOURNAL =. 1987 , NUMBER =. doi:10.1007/BF01404671 , URL =

1987 doi
[23]

Blaine and Thurston, William , TITLE =

Gromov, Mikhail and Lawson, Jr., H. Blaine and Thurston, William , TITLE =. Inst. Hautes \'. 1988 , PAGES =

1988
[24]

Kapovich, Michael , TITLE =. J. Differential Geom. , FJOURNAL =. 1993 , NUMBER =

1993
[25]

Groups Geom

Giralt, Anne , TITLE =. Groups Geom. Dyn. , FJOURNAL =. 2017 , NUMBER =. doi:10.4171/GGD/401 , URL =

2017 doi
[26]

Dani, S. G. and Margulis, G. A. , title =. I. M. Gelfand Seminar. Part 1: Papers of the Gelfand seminar in functional analysis held at Moscow University, Russia, September 1993 , isbn =. 1993 , publisher =

1993
[27]

Ratner, Marina , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1991 , NUMBER =. doi:10.2307/2944357 , URL =

1991 doi
[28]

Eskin, Alex and Mozes, Shahar and Shah, Nimish , TITLE =. Geom. Funct. Anal. , FJOURNAL =. 1997 , NUMBER =. doi:10.1007/PL00001616 , URL =

1997 doi
[29]

Eskin, Alex and Mozes, Shahar and Shah, Nimish , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1996 , NUMBER =. doi:10.2307/2118644 , URL =

1996 doi
[30]

Sullivan, Dennis , title =. J. Differ. Geom. , issn =. 1987 , language =. doi:10.4310/jdg/1214440979 , keywords =

1987 doi
[31]

Buser, Peter , title =. Ann. Sci. 1982 , language =. doi:10.24033/asens.1426 , keywords =

1982 doi
[32]

Dani, S. G. , title =. Ergodic Theory Dyn. Syst. , issn =. 1984 , language =. doi:10.1017/S0143385700002248 , keywords =

1984 doi
[33]

Conformal geometry (

Kulkarni, Ravi S , TITLE =. Conformal geometry (. 1988 , MRCLASS =

1988
[34]

1988 , PAGES =

Maskit, Bernard , TITLE =. 1988 , PAGES =

1988
[35]

Ergodic Theory Dyn

Benoist, Yves and Oh, Hee , title =. Ergodic Theory Dyn. Syst. , issn =. 2022 , language =. doi:10.1017/etds.2021.19 , keywords =

2022 doi
[36]

and Mohammadi, Amir and Oh, Hee , title =

McMullen, Curtis T. and Mohammadi, Amir and Oh, Hee , title =. Invent. Math. , issn =. 2017 , language =. doi:10.1007/s00222-016-0711-3 , keywords =

2017 doi
[37]

Linear Algebra Appl

Koberda, Thomas and Luo, Feng and Sun, Hongbin , TITLE =. Linear Algebra Appl. , FJOURNAL =. 2019 , PAGES =. doi:10.1016/j.laa.2019.07.023 , URL =

2019 doi
[38]

Kleiner, Bruce and Leeb, Bernhard , TITLE =. Invent. Math. , FJOURNAL =. 2006 , NUMBER =. doi:10.1007/s00222-005-0471-y , URL =

2006 doi
[39]

Marden, Albert , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1974 , PAGES =. doi:10.2307/1971059 , URL =

1974 doi
[40]

Acta Math

Sullivan, Dennis , TITLE =. Acta Math. , FJOURNAL =. 1985 , NUMBER =. doi:10.1007/BF02392543 , URL =

1985 doi
[41]

Breuillard, Emmanuel and Gelander, Tsachik , TITLE =. Invent. Math. , FJOURNAL =. 2008 , NUMBER =. doi:10.1007/s00222-007-0101-y , URL =

2008 doi
[42]

, TITLE = "

Massey, William S. , TITLE = ". 1991 , PAGES =

1991
[43]

Groups, Geometry, and Dynamics , volume=

Arcs on punctured disks intersecting at most twice with endpoints on the boundary , author=. Groups, Geometry, and Dynamics , volume=
[44]

Borel, Armand , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1960 , PAGES =. doi:10.2307/1970150 , URL =

1960 doi
[45]

Culler, Marc , TITLE =. Adv. in Math. , FJOURNAL =. 1986 , NUMBER =. doi:10.1016/0001-8708(86)90037-X , URL =

1986 doi
[46]

Lubotzky, Alexander and Venkataramana, Tyakal Nanjundiah , TITLE =. Alg. Number Th. , FJOURNAL =. 2019 , NUMBER =. doi:10.2140/ant.2019.13.1281 , URL =

2019 doi
[47]

Breuillard, Emmanuel and Gelander, Tsachik and Souto, Juan and Storm, Peter , TITLE =. Geom. Topol. , FJOURNAL =. 2006 , PAGES =. doi:10.2140/gt.2006.10.1373 , URL =

2006 doi
[48]

2014 , publisher=

Thin groups and superstrong approximation , author=. 2014 , publisher=

2014
[49]

Duke Math

Ratner, Marina , TITLE =. Duke Math. J. , FJOURNAL =. 1991 , NUMBER =. doi:10.1215/S0012-7094-91-06311-8 , URL =

1991 doi
[50]

Israel J

Oh, Hee , TITLE =. Israel J. Math. , FJOURNAL =. 1999 , PAGES =. doi:10.1007/BF02808188 , URL =

1999 doi
[51]

Sullivan, Dennis , title =. Publ. Math., Inst. Hautes. 1979 , language =. doi:10.1007/BF02684773 , keywords =

1979 doi
[52]

Acta Math

Sullivan, Dennis , title =. Acta Math. , issn =. 1984 , language =. doi:10.1007/BF02392379 , keywords =

1984 doi
[53]

Oh, Hee , TITLE =. C. R. Acad. Sci. Paris S\'. 1996 , NUMBER =

1996
[54]

Computational

Kapovich, Michael and Detinko, Alla and Kontorovich, Alex , TITLE =. Computational. [2023] 2023 , MRCLASS =. doi:10.1090/conm/783/15737 , URL =

2023 doi
[55]

Thin groups and superstrong approximation , SERIES =

Sarnak, Peter , TITLE =. Thin groups and superstrong approximation , SERIES =. 2014 , MRCLASS =

2014
[56]

Douba, Sami , TITLE =. Bull. Lond. Math. Soc. , FJOURNAL =. 2022 , NUMBER =

2022
[57]

Mostow, G. D. and Tamagawa, T. , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1962 , PAGES =. doi:10.2307/1970368 , URL =

1962 doi
[58]

Borel, Armand and Harish-Chandra , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1962 , PAGES =. doi:10.2307/1970210 , URL =

1962 doi
[59]

Groups, Geometry, and Dynamics , volume=

Superrigidity in infinite dimension and finite rank via harmonic maps , author=. Groups, Geometry, and Dynamics , volume=
[60]

Essays in group theory , SERIES =

Gromov, Mikhail , TITLE =. Essays in group theory , SERIES =. 1987 , MRCLASS =. doi:10.1007/978-1-4613-9586-7\_3 , URL =

1987 doi
[61]

Espaces m\'

Ghys, \'. Espaces m\'. Sur les groupes hyperboliques d'apr\`es. 1990 , MRCLASS =. doi:10.1007/978-1-4684-9167-8\_2 , URL =

1990 doi
[62]

, TITLE =

Bourbaki, N. , TITLE =. 1968 , PAGES =

1968
[63]

Cubulated hyperbolic groups admit

Douba, Sami and Fl. Cubulated hyperbolic groups admit. arXiv preprint arXiv:2309.03695 , year=
[64]

arXiv preprint arXiv:2309.16051 , year=

Systoles of hyperbolic hybrids , author=. arXiv preprint arXiv:2309.16051 , year=

arXiv
[65]

Shalom, Yehuda , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 1999 , NUMBER =. doi:10.1090/S0002-9947-99-02363-6 , URL =

1999 doi
[66]

Les groupes hyperboliques , NOTE =

Ghys, \'. Les groupes hyperboliques , NOTE =. Ast\'. 1990 , PAGES =

1990
[67]

Journal of Pure and Applied Algebra , volume=

Linear representations of certain amalgamated products , author=. Journal of Pure and Applied Algebra , volume=. 1979 , publisher=

1979
[68]

Semisimple actions of mapping class groups on CAT(0) spaces

Bridson, Martin R , journal=. Semisimple actions of mapping class groups on CAT(0) spaces. 2010 , publisher=

2010
[69]

Topics in Geometric Group Theory

de La Harpe, Pierre , year=. Topics in Geometric Group Theory
[70]

On the first cohomology group of discrete groups with property (T)

Wang, Shu Ping , journal=. On the first cohomology group of discrete groups with property (T)
[71]

Linearisability of the Fundamental Groups of 3 -Manifolds , author=
[72]

2 -Systems of arcs on spheres with prescribed endpoints

Douba, Sami , journal=. 2 -Systems of arcs on spheres with prescribed endpoints. 2021 , publisher=

2021
[73]

Geometric and Functional Analysis , volume=

Arcs intersecting at most once , author=. Geometric and Functional Analysis , volume=. 2015 , publisher=

2015
[74]

Journal of the American Mathematical Society , volume=

Mixed 3-manifolds are virtually special , author=. Journal of the American Mathematical Society , volume=
[75]

https://math.stackexchange.com/q/792331 , URL =

Showing that every finitely presented group has a 4 -manifold with it as its fundamental group , AUTHOR =. https://math.stackexchange.com/q/792331 , URL =
[76]

Wise, D. T. , TITLE =. [2021] 2021 , PAGES =

2021
[77]

On regular subgroups of SL_3( R )

Douba, Sami and Tsouvalas, Konstantinos , journal=. On regular subgroups of SL_3( R )
[78]

3-Manifold Groups

Aschenbrenner, Matthias and Friedl, Stefan and Wilton, Henry , volume=. 3-Manifold Groups. 2015 , publisher=

2015
[79]

Des espaces de Hadamard sym

Duchesne, Bruno , year=. Des espaces de Hadamard sym
[80]

Lee, Gye-Seon and Marquis, Ludovic , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2019 , NUMBER =. doi:10.1090/tran/7530 , URL =

2019 doi
[81]

Hyperbolic

Moussong, Gabor , year=. Hyperbolic
[82]

arXiv preprint arXiv:2210.17549 , year=

Anosov groups that are indiscrete in rank one , author=. arXiv preprint arXiv:2210.17549 , year=

arXiv
[83]

Robust quasi-isometric embeddings inapproximable by

Tsouvalas, Konstantinos , journal=. Robust quasi-isometric embeddings inapproximable by
[84]

arXiv preprint arXiv:2112.05574 , year=

Linearity and indiscreteness of amalgamated products of hyperbolic groups , author=. arXiv preprint arXiv:2112.05574 , year=

arXiv
[85]

Brooks, Robert , TITLE =. Invent. Math. , FJOURNAL =. 1986 , NUMBER =. doi:10.1007/BF01389263 , URL =

1986 doi
[86]

2022 , school=

Unipotents and graph manifold groups , author=. 2022 , school=

2022
[87]

Representations of Compact Lie Groups

Br. Representations of Compact Lie Groups. 1985 , publisher=

1985
[88]

Caprace, Pierre-Emmanuel and Monod, Nicolas , TITLE =. J. Topol. , FJOURNAL =. 2009 , NUMBER =. doi:10.1112/jtopol/jtp026 , URL =

2009 doi
[89]

2013 , publisher=

3-manifold groups are virtually residually p , author=. 2013 , publisher=

2013
[90]

Metric Spaces of Non-Positive Curvature

Bridson, Martin R and Haefliger, Andr. Metric Spaces of Non-Positive Curvature. 1999 , publisher=

1999
[91]

3-manifolds with(out) metrics of nonpositive curvature , school =

Leeb, Bernhard , year =. 3-manifolds with(out) metrics of nonpositive curvature , school =
[92]

arXiv preprint arXiv:1107.2019 , year=

Rigidity of high dimensional graph manifolds , author=. arXiv preprint arXiv:1107.2019 , year=

2019 arXiv
[93]

Sageev, Michah , TITLE =. Proc. London Math. Soc. (3) , FJOURNAL =. 1995 , NUMBER =. doi:10.1112/plms/s3-71.3.585 , URL =

1995 doi
[94]

Sageev, Michah , TITLE =. J. Algebra , FJOURNAL =. 1997 , NUMBER =. doi:10.1006/jabr.1996.6884 , URL =

1997 doi
[95]

Geometric topology (

Luecke, John and Wu, Ying-Qing , TITLE =. Geometric topology (. 1997 , MRCLASS =. doi:10.1090/amsip/002.1/05 , URL =

1997 doi
[96]

Bestvina, Mladen and Mess, Geoffrey , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 1991 , NUMBER =. doi:10.2307/2939264 , URL =

1991 doi
[97]

Bowditch, B. H. and Mess, G. , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 1994 , NUMBER =. doi:10.2307/2154722 , URL =

1994 doi
[98]

Benz\'ecri, Jean-Paul , TITLE =. Bull. Soc. Math. France , FJOURNAL =. 1960 , PAGES =

1960
[99]

Analysis on homogeneous spaces and representation theory of

Benoist, Yves , TITLE =. Analysis on homogeneous spaces and representation theory of. 2000 , MRCLASS =. doi:10.2969/aspm/02610033 , URL =

2000 doi
[100]

Klein, Felix , TITLE =. Math. Ann. , FJOURNAL =. 1883 , NUMBER =. doi:10.1007/BF01442920 , URL =

doi
[101]

Bonk, Mario and Kleiner, Bruce and Merenkov, Sergei , title =. Am. J. Math. , issn =. 2009 , language =. doi:10.1353/ajm.0.0045 , keywords =

2009 doi
[102]

Frigerio, Roberto , TITLE =. Geom. Dedicata , FJOURNAL =. 2006 , PAGES =. doi:10.1007/s10711-005-9028-x , URL =

2006 doi
[103]

Topology , FJOURNAL =

Miyamoto, Yosuke , TITLE =. Topology , FJOURNAL =. 1994 , NUMBER =. doi:10.1016/0040-9383(94)90001-9 , URL =

1994 doi
[104]

and Svetlov, Pavel V , TITLE =

Buyalo, Sergei V. and Svetlov, Pavel V , TITLE =. Algebra i Analiz , FJOURNAL =. 2004 , NUMBER =. doi:10.1090/S1061-0022-05-00852-6 , URL =

2004 doi
[105]

Wise, D. T. , TITLE =. Bull. Lond. Math. Soc. , FJOURNAL =. 2011 , NUMBER =. doi:10.1112/blms/bdr023 , URL =

2011 doi
[106]

Kapovich, Michael , TITLE =. Geom. Topol. , FJOURNAL =. 2013 , NUMBER =. doi:10.2140/gt.2013.17.39 , URL =

2013 doi
[107]

Kapovich, Michael and Potyagailo, Leonid and Vinberg, Ernest , TITLE =. The. 2008 , MRCLASS =. doi:10.2140/gtm.2008.14.335 , URL =

2008 doi
[108]

Malcev, Anatolii I , TITLE =. Amer. Math. Soc. Transl. (2) , FJOURNAL =. 1956 , PAGES =

1956
[109]

Swan, Richard G , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 1967 , PAGES =. doi:10.2307/2035503 , URL =

1967 doi
[110]

Potyagailo, Leonid and Vinberg, Ernest , TITLE =. Comment. Math. Helv. , FJOURNAL =. 2005 , NUMBER =. doi:10.4171/CMH/4 , URL =

2005 doi
[111]

Osajda, Damian , TITLE =. Comment. Math. Helv. , FJOURNAL =. 2013 , NUMBER =. doi:10.4171/CMH/288 , URL =

2013 doi
[112]

Uspekhi Mat

Vinberg, \`Ernest B , TITLE =. Uspekhi Mat. Nauk , FJOURNAL =. 1985 , NUMBER =

1985
[113]

Vinberg, \`Ernest B , TITLE =. Izv. Akad. Nauk SSSR Ser. Mat. , FJOURNAL =. 1971 , PAGES =

1971
[114]

Combinatorial Group Theory and Topology.(AM-111), Volume 111 , pages=

Residual finiteness for 3-manifolds , author=. Combinatorial Group Theory and Topology.(AM-111), Volume 111 , pages=. 2016 , publisher=

2016
[115]

The virtual Haken conjecture

Agol, Ian and Groves, Daniel and Manning, Jason , journal=. The virtual Haken conjecture
[116]

On some convex cocompact groups in real hyperbolic space , JOURNAL =

Desgroseilliers, Marc and Haglund, Fr\'. On some convex cocompact groups in real hyperbolic space , JOURNAL =. 2013 , NUMBER =. doi:10.2140/gt.2013.17.2431 , URL =

2013 doi
[117]

Annals of Mathematics , pages=

Immersing almost geodesic surfaces in a closed hyperbolic three manifold , author=. Annals of Mathematics , pages=. 2012 , publisher=

2012
[118]

Tangent bundles of hyperbolic spaces and proper affine actions on L^ p spaces

Chatterji, Indira and Dahmani, Fran. Tangent bundles of hyperbolic spaces and proper affine actions on L^ p spaces. arXiv preprint arXiv:1901.07462 , year=

1901 arXiv
[119]

and Raghunathan, M

Millson, John J. and Raghunathan, M. S. , TITLE =. Geometry and analysis , PAGES =. 1980 , MRCLASS =

1980
[120]

, TITLE =

Millson, John J. , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1976 , NUMBER =. doi:10.2307/1971046 , URL =

1976 doi
[121]

Electronic Research Announcements , volume=

Research announcement: the structure of groups with a quasiconvex hierarchy , author=. Electronic Research Announcements , volume=. 2009 , publisher=

2009
[122]

, TITLE =

Johnson, Dennis and Millson, John J. , TITLE =. Discrete groups in geometry and analysis (. 1987 , MRCLASS =. doi:10.1007/978-1-4899-6664-3\_3 , URL =

1987 doi
[123]

Wise, D. T. , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2014 , NUMBER =. doi:10.1090/S0002-9947-2014-06065-0 , URL =

2014 doi
[124]

, TITLE =

Woodhouse, Daniel J. , TITLE =. Groups Geom. Dyn. , FJOURNAL =. 2018 , NUMBER =. doi:10.4171/GGD/452 , URL =

2018 doi
[125]

Markovic, Vladimir , TITLE =. Geom. Funct. Anal. , FJOURNAL =. 2013 , NUMBER =. doi:10.1007/s00039-013-0228-5 , URL =

2013 doi
[126]

American Journal of Mathematics , volume=

A boundary criterion for cubulation , author=. American Journal of Mathematics , volume=. 2012 , publisher=

2012
[127]

and Przytycki, Piotr , TITLE =

Hagen, Mark F. and Przytycki, Piotr , TITLE =. Israel J. Math. , FJOURNAL =. 2015 , NUMBER =. doi:10.1007/s11856-015-1177-5 , URL =

2015 doi
[128]

A combination theorem for special cube complexes , JOURNAL =

Haglund, Fr\'. A combination theorem for special cube complexes , JOURNAL =. 2012 , NUMBER =. doi:10.4007/annals.2012.176.3.2 , URL =

2012 doi
[129]

Hyperbolic groups with planar boundaries , JOURNAL =

Ha\". Hyperbolic groups with planar boundaries , JOURNAL =. 2015 , NUMBER =. doi:10.1007/s00222-014-0552-x , URL =

2015 doi
[130]

Hyperplane sections in arithmetic hyperbolic manifolds , JOURNAL =

Bergeron, Nicolas and Haglund, Fr\'. Hyperplane sections in arithmetic hyperbolic manifolds , JOURNAL =. 2011 , NUMBER =. doi:10.1112/jlms/jdq082 , URL =

2011 doi
[131]

Prospects in mathematics (

Serre, Jean-Pierre , TITLE =. Prospects in mathematics (. 1971 , MRCLASS =

1971
[132]

Coxeter groups are virtually special , JOURNAL =

Haglund, Fr\'. Coxeter groups are virtually special , JOURNAL =. 2010 , NUMBER =. doi:10.1016/j.aim.2010.01.011 , URL =

2010 doi
[133]

, author=

On linear and residual properties of graph products. , author=. Michigan Mathematical Journal , volume=. 1999 , publisher=

1999
[134]

Journal of Topology , volume=

Graph manifolds with boundary are virtually special , author=. Journal of Topology , volume=. 2014 , publisher=

2014
[135]

2015 , PAGES =

Aschenbrenner, Matthias and Friedl, Stefan and Wilton, Henry , TITLE =. 2015 , PAGES =. doi:10.4171/154 , URL =

2015 doi
[136]

Farrell, F. T. and Jones, L. E. , TITLE =. Proc. Nat. Acad. Sci. U.S.A. , FJOURNAL =. 1989 , NUMBER =. doi:10.1073/pnas.86.10.3461 , URL =

1989 doi
[137]

Geometric and Functional Analysis , volume=

Special cube complexes , author=. Geometric and Functional Analysis , volume=. 2008 , publisher=

2008
[138]

Inventiones mathematicae , volume=

3-manifolds with (out) metrics of nonpositive curvature , author=. Inventiones mathematicae , volume=. 1995 , publisher=

1995
[139]

Pure Appl

Stover, Matthew and Toledo, Domingo , TITLE =. Pure Appl. Math. Q. , FJOURNAL =. 2022 , NUMBER =

2022
[140]

2015 , PAGES =

Morris, Dave Witte , TITLE =. 2015 , PAGES =

2015
[141]

Groups Complexity Cryptology , volume=

Free by cyclic groups and linear groups with restricted unipotent elements , author=. Groups Complexity Cryptology , volume=. 2017 , publisher=

2017
[142]

, author=

Aspects of non positive curvature for linear groups with no infinite order unipotents. , author=. Groups, Geometry, and Dynamics , volume=. 2019 , publisher=

2019
[143]

Stalder, Yves and Valette, Alain , TITLE =. Geom. Dedicata , FJOURNAL =. 2007 , PAGES =. doi:10.1007/s10711-006-9119-3 , URL =

2007 doi
[144]

, TITLE =

Davis, Michael W. , TITLE =. Introduction to modern mathematics , SERIES =. 2015 , MRCLASS =

2015
[145]

Davis, Tara C and Olshanskii, Alexander Y , TITLE =. J. Pure Appl. Algebra , FJOURNAL =. 2011 , NUMBER =. doi:10.1016/j.jpaa.2011.04.018 , URL =

2011 doi
[146]

Eberlein, Patrick , TITLE =. Math. Ann. , FJOURNAL =. 1982 , NUMBER =. doi:10.1007/BF01475751 , URL =

1982 doi
[147]

1996 , PAGES =

Eberlein, Patrick B , TITLE =. 1996 , PAGES =

1996
[148]

, TITLE =

Prasad, Gopal and Rapinchuk, Andrei S. , TITLE =. Internat. Math. Res. Notices , FJOURNAL =. 2001 , NUMBER =. doi:10.1155/S1073792801000587 , URL =

2001 doi
[149]

Margulis, G. A. , TITLE =. 1991 , PAGES =. doi:10.1007/978-3-642-51445-6 , URL =

1991 doi
[150]

1998 , PAGES =

Brown, Kenneth S , TITLE =. 1998 , PAGES =

1998
[151]

Conner, Gregory R , TITLE =. J. Group Theory , FJOURNAL =. 2000 , NUMBER =. doi:10.1515/jgth.2000.007 , URL =

2000 doi
[152]

1972 , PAGES =

Raghunathan, Madabusi S , TITLE =. 1972 , PAGES =

1972
[153]

Groups Geom

Kapovich, Michael and Leeb, Bernhard , TITLE =. Groups Geom. Dyn. , FJOURNAL =. 2023 , NUMBER =. doi:10.4171/ggd/723 , URL =

2023 doi
[154]

Zhu, Feng , TITLE =. Ann. Inst. Fourier (Grenoble) , FJOURNAL =. 2021 , NUMBER =. doi:10.5802/aif.3449 , URL =

2021 doi
[155]

Proceedings of the

Wienhard, Anna , TITLE =. Proceedings of the. 2018 , ISBN =

2018
[156]

Anosov flows, surface groups and curves in projective space , JOURNAL =

Labourie, Fran. Anosov flows, surface groups and curves in projective space , JOURNAL =. 2006 , NUMBER =. doi:10.1007/s00222-005-0487-3 , URL =

2006 doi
[157]

Ast\'erisque , FJOURNAL =

Breuillard, Emmanuel , TITLE =. Ast\'erisque , FJOURNAL =. 2010 , PAGES =

2010
[158]

and Lindenstrauss, E

Einsiedler, M. and Lindenstrauss, E. , TITLE =. Homogeneous flows, moduli spaces and arithmetic , SERIES =. 2010 , ISBN =

2010
[159]

Margulis, G. A. , TITLE =. Bull. Soc. Math. France , FJOURNAL =. 1997 , NUMBER =

1997
[160]

Funkcional

Kazhdan, David , TITLE =. Funkcional. Anal. i Prilo zen. , FJOURNAL =. 1967 , PAGES =

1967
[161]

Kazhdan, David , TITLE =. J. Analyse Math. , FJOURNAL =. 1977 , PAGES =. doi:10.1007/bf02803582 , URL =

1977 doi
[162]

Kielak, Dawid , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2020 , NUMBER =. doi:10.1090/jams/936 , URL =

2020 doi
[163]

Ergodic Theory Dynam

Guivarc'h, Yves , TITLE =. Ergodic Theory Dynam. Systems , FJOURNAL =. 1990 , NUMBER =. doi:10.1017/S0143385700005708 , URL =

1990 doi
[164]

Convexes divisibles (d'apr\`es

Quint, Jean-Fran. Convexes divisibles (d'apr\`es. Ast\'erisque , FJOURNAL =. 2010 , PAGES =

2010
[165]

, TITLE =

Quint, J.-F. , TITLE =. Geom. Dedicata , FJOURNAL =. 2005 , PAGES =. doi:10.1007/s10711-005-0122-x , URL =

2005 doi
[166]

Hyperbolic

Januszkiewicz, Tadeusz and \'. Hyperbolic. Comment. Math. Helv. , FJOURNAL =. 2003 , NUMBER =. doi:10.1007/s00014-003-0763-z , URL =

2003 doi
[167]

Arithmeticity of discrete subgroups containing horospherical lattices , JOURNAL =

Benoist, Yves and Miquel, S\'. Arithmeticity of discrete subgroups containing horospherical lattices , JOURNAL =. 2020 , NUMBER =. doi:10.1215/00127094-2019-0082 , URL =

2020 doi
[168]

Prasad, Gopal and Raghunathan, M. S. , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1972 , PAGES =. doi:10.2307/1970790 , URL =

1972 doi
[169]

arXiv preprint arXiv:1805.10189 , year=

Surface groups in uniform lattices of some semi-simple groups , author=. arXiv preprint arXiv:1805.10189 , year=

arXiv
[170]

Lubotzky, Alexander and Mozes, Shahar and Raghunathan, Madabusi S , TITLE =. Inst. Hautes \'. 2000 , PAGES =

2000
[171]

Benoist, Yves and de la Harpe, Pierre , TITLE =. Compos. Math. , FJOURNAL =. 2004 , NUMBER =. doi:10.1112/S0010437X04000338 , URL =

2004 doi
[172]

Preprint , year=

A POLYHEDRAL APPROACH TO THE ARITHMETIC AND GEOMETRY OF HYPERBOLIC LINK COMPLEMENTS , author=. Preprint , year=
[173]

Benoist, Yves , TITLE =. Invent. Math. , FJOURNAL =. 2006 , NUMBER =. doi:10.1007/s00222-005-0478-4 , URL =

2006 doi
[174]

Lectures on representations of surface groups , SERIES =

Labourie, Fran. Lectures on representations of surface groups , SERIES =. 2013 , PAGES =. doi:10.4171/127 , URL =

2013 doi
[175]

Benoist, Yves , TITLE =. G\'. 2009 , MRCLASS =

2009
[176]

arXiv preprint arXiv:2206.01123 , year=

Zariski-dense surface groups in non-uniform lattices of split real Lie groups , author=. arXiv preprint arXiv:2206.01123 , year=
[177]

A hyperbolic lattice in each dimension with

Douba, Sami , journal=. A hyperbolic lattice in each dimension with
[178]

Experiment

Ratcliffe, John G and Tschantz, Steven T , TITLE =. Experiment. Math. , FJOURNAL =. 2000 , NUMBER =

2000
[179]

Prokhorov, Mikhail N , TITLE =. Izv. Akad. Nauk SSSR Ser. Mat. , FJOURNAL =. 1986 , NUMBER =

1986
[180]

Long, D. D. and Thistlethwaite, M. B. , TITLE =. Geom. Topol. , FJOURNAL =. 2024 , NUMBER =. doi:10.2140/gt.2024.28.1153 , URL =

2024 doi
[181]

Long, D. D. and Thistlethwaite, M. B. , TITLE =. Exp. Math. , FJOURNAL =. 2018 , NUMBER =. doi:10.1080/10586458.2016.1226992 , URL =

2018 doi
[182]

Illinois J

Long, Darren D and Reid, Alan W , TITLE =. Illinois J. Math. , FJOURNAL =. 2016 , NUMBER =

2016
[183]

Groupes de surface dans les r

Kassel, Fanny , journal=. Groupes de surface dans les r
[184]

1977 , PAGES =

Serre, Jean-Pierre , TITLE =. 1977 , PAGES =

1977
[185]

Duke Math

Farb, Benson and Lubotzky, Alexander and Minsky, Yair , TITLE =. Duke Math. J. , FJOURNAL =. 2001 , NUMBER =. doi:10.1215/S0012-7094-01-10636-4 , URL =

2001 doi
[186]

Mark, Alice and Paupert, Julien , TITLE =. Algebr. Geom. Topol. , FJOURNAL =. 2022 , NUMBER =. doi:10.2140/agt.2022.22.3577 , URL =

2022 doi
[187]

, TITLE =

Rapinchuk, A. , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 1999 , NUMBER =. doi:10.1090/S0002-9939-99-04696-1 , URL =

1999 doi
[188]

Kotowski, Marcin and Kotowski, Micha , TITLE =. J. Lond. Math. Soc. (2) , FJOURNAL =. 2013 , NUMBER =. doi:10.1112/jlms/jdt024 , URL =

2013 doi
[189]

Caprace, Pierre-Emmanuel and Conder, Marston and Kaluba, Marek and Witzel, Stefan , TITLE =. J. Lond. Math. Soc. (2) , FJOURNAL =. 2022 , NUMBER =. doi:10.1112/jlms.12668 , URL =

2022 doi
[190]

Zuk, Andrzej , TITLE =. Geom. Funct. Anal. , FJOURNAL =. 2003 , NUMBER =. doi:10.1007/s00039-003-0425-8 , URL =

2003 doi
[191]

Zuk, Andrzej , TITLE =. C. R. Acad. Sci. Paris S\'er. I Math. , FJOURNAL =. 1996 , NUMBER =

1996
[192]

Gersten, Stephen M , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 1994 , NUMBER =. doi:10.2307/2161207 , URL =

1994 doi
[193]

Bergeron, Nicolas and Clozel, Laurent , TITLE =. Invent. Math. , FJOURNAL =. 2013 , NUMBER =. doi:10.1007/s00222-012-0415-2 , URL =

2013 doi
[194]

Baumslag, Gilbert , TITLE =. Math. Z. , FJOURNAL =. 1962 , PAGES =. doi:10.1007/BF01195185 , URL =

1962 doi
[195]

Marquis, Ludovic , TITLE =. Geom. Dedicata , FJOURNAL =. 2010 , PAGES =. doi:10.1007/s10711-009-9438-2 , URL =

2010 doi
[196]

Vershik, A. M. and Gelfand, I. M. and Graev, M. I. , TITLE =. Funkcional. Anal. i Prilo zen. , FJOURNAL =. 1974 , NUMBER =

1974
[197]

Guichard, Olivier and Wienhard, Anna , TITLE =. Invent. Math. , FJOURNAL =. 2012 , NUMBER =. doi:10.1007/s00222-012-0382-7 , URL =

2012 doi
[198]

Mila, Olivier , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2021 , NUMBER =. doi:10.1093/imrn/rnaa114 , URL =

2021 doi
[199]

, TITLE =

Benoist, Y. , TITLE =. Geom. Funct. Anal. , FJOURNAL =. 1997 , NUMBER =. doi:10.1007/PL00001613 , URL =

1997 doi
[200]

Lie groups and their representations (

Kostant, Bertram , TITLE =. Lie groups and their representations (. 1975 , MRCLASS =

1975
[201]

Groups Geom

Koivisto, Juhani and Kyed, David and Raum, Sven , TITLE =. Groups Geom. Dyn. , FJOURNAL =. 2021 , NUMBER =. doi:10.4171/ggd/597 , URL =

2021 doi
[202]

, TITLE =

Gromov, M. , TITLE =. Geometric group theory,. 1993 , MRCLASS =

1993
[203]

, TITLE =

Ollivier, Yann and Wise, Daniel T. , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2011 , NUMBER =. doi:10.1090/S0002-9947-2011-05197-4 , URL =

2011 doi
[204]

Kozma, Gady and Lubotzky, Alexander , TITLE =. Bull. Math. Sci. , FJOURNAL =. 2019 , NUMBER =. doi:10.1142/S1664360719500164 , URL =

2019 doi
[205]

Deraux, Martin , TITLE =. Geom. Topol. , FJOURNAL =. 2016 , NUMBER =. doi:10.2140/gt.2016.20.3571 , URL =

2016 doi
[206]

Kulkarni, Ravi S , TITLE =. J. Differential Geometry , FJOURNAL =. 1978 , NUMBER =

1978
[207]

Deformation spaces of

Choi, Suhyoung and Lee, Gye-Seon and Marquis, Ludovic , journal=. Deformation spaces of. 2021 , publisher=

2021
[208]

Ballas, Samuel A and Long, Darren D , TITLE =. Algebr. Geom. Topol. , FJOURNAL =. 2020 , NUMBER =. doi:10.2140/agt.2020.20.2071 , URL =

2020 doi
[209]

, TITLE =

Ballas, Samuel A. , TITLE =. Pacific J. Math. , FJOURNAL =. 2020 , NUMBER =. doi:10.2140/pjm.2020.309.257 , URL =

2020 doi
[210]

Benoist, Yves , TITLE =. Invent. Math. , FJOURNAL =. 2000 , NUMBER =. doi:10.1007/PL00005789 , URL =

2000 doi
[211]

Kapovich, Michael , TITLE =. Geom. Topol. , FJOURNAL =. 2007 , PAGES =. doi:10.2140/gt.2007.11.1777 , URL =

2007 doi
[212]

Ballas, Samuel A and Danciger, Jeffrey and Lee, Gye-Seon , TITLE =. Geom. Topol. , FJOURNAL =. 2018 , NUMBER =. doi:10.2140/gt.2018.22.1593 , URL =

2018 doi
[213]

Acta Math

Namazi, Hossein and Souto, Juan , TITLE =. Acta Math. , FJOURNAL =. 2012 , NUMBER =. doi:10.1007/s11511-012-0088-0 , URL =

2012 doi
[214]

Ohshika, Ken'ichi , TITLE =. Geom. Topol. , FJOURNAL =. 2011 , NUMBER =. doi:10.2140/gt.2011.15.827 , URL =

2011 doi
[215]

Bergeron, Nicolas , TITLE =. Enseign. Math. (2) , FJOURNAL =. 2000 , NUMBER =

2000
[216]

Ast\'erisque , FJOURNAL =

Bergeron, Nicolas , TITLE =. Ast\'erisque , FJOURNAL =. 2015 , PAGES =

2015
[217]

Israel J

Bergeron, Nicolas and Clozel, Laurent , TITLE =. Israel J. Math. , FJOURNAL =. 2017 , NUMBER =. doi:10.1007/s11856-017-1593-9 , URL =

2017 doi
[218]

A novel witness to incoherence of SL_5( Z )

Douba, Sami , journal=. A novel witness to incoherence of SL_5( Z )
[219]

Agol, Ian , TITLE =. J. Topol. , FJOURNAL =. 2008 , NUMBER =. doi:10.1112/jtopol/jtn003 , URL =

2008 doi
[220]

Choi, Kanghyun and Choi, Suhyoung , TITLE =. Geom. Dedicata , FJOURNAL =. 2015 , PAGES =. doi:10.1007/s10711-014-9949-3 , URL =

2015 doi
[221]

Algebraic groups and arithmetic , PAGES =

Benoist, Yves , TITLE =. Algebraic groups and arithmetic , PAGES =. 2004 , MRCLASS =

2004
[222]

, TITLE =

Kochloukova, Dessislava H. , TITLE =. Internat. J. Algebra Comput. , FJOURNAL =. 2016 , NUMBER =. doi:10.1142/S0218196716500235 , URL =

2016 doi
[223]

arXiv preprint arXiv:2105.06897 , year=

Subspace stabilisers in hyperbolic lattices , author=. arXiv preprint arXiv:2105.06897 , year=
[224]

Py, Pierre , TITLE =. Math. Proc. Cambridge Philos. Soc. , FJOURNAL =. 2013 , NUMBER =. doi:10.1017/S0305004113000534 , URL =

2013 doi
[225]

arXiv preprint arXiv:1505.03643 , year=

Totally geodesic spectra of quaternionic hyperbolic orbifolds , author=. arXiv preprint arXiv:1505.03643 , year=

arXiv
[226]

Corlette, Kevin , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1992 , NUMBER =. doi:10.2307/2946567 , URL =

1992 doi
[227]

Forum Math

Emery, Vincent and Kim, Inkang , TITLE =. Forum Math. Sigma , FJOURNAL =. 2022 , PAGES =. doi:10.1017/fms.2022.43 , URL =

2022 doi
[228]

Venkataramana, T. N. , TITLE =. Internat. Math. Res. Notices , FJOURNAL =. 1999 , NUMBER =. doi:10.1155/S1073792899000422 , URL =

1999 doi
[229]

Kapovich, Michael and Kleiner, Bruce , title =. Ann. Sci. 2000 , language =. doi:10.1016/S0012-9593(00)01049-1 , keywords =

2000 doi
[230]

Groups Geom

Bourdon, Marc and Kleiner, Bruce , TITLE =. Groups Geom. Dyn. , FJOURNAL =. 2013 , NUMBER =. doi:10.4171/GGD/177 , URL =

2013 doi
[231]

Infinite groups: geometric, combinatorial and dynamical aspects , SERIES =

Delzant, Thomas and Gromov, Mikhail , TITLE =. Infinite groups: geometric, combinatorial and dynamical aspects , SERIES =. 2005 , MRCLASS =. doi:10.1007/3-7643-7447-0\_3 , URL =

2005 doi
[232]

Gromov, Mikhail and Schoen, Richard , TITLE =. Inst. Hautes \'. 1992 , PAGES =

1992
[233]

2015 , eprint=

Introduction to Arithmetic Groups , author=. 2015 , eprint=

2015
[234]

Wise, D. T. , TITLE =. Geom. Funct. Anal. , FJOURNAL =. 2004 , NUMBER =. doi:10.1007/s00039-004-0454-y , URL =

2004 doi
[235]

Hall, Jr, Marshall , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 1949 , PAGES =. doi:10.2307/1990483 , URL =

1949 doi
[236]

Basmajian, Ara , TITLE =. Invent. Math. , FJOURNAL =. 1994 , NUMBER =. doi:10.1007/BF01232240 , URL =

1994 doi
[237]

and Sarnak, P

Burger, M. and Sarnak, P. , TITLE =. Invent. Math. , FJOURNAL =. 1991 , NUMBER =. doi:10.1007/BF01243900 , URL =

1991 doi
[238]

Long, D. D. , TITLE =. Bull. London Math. Soc. , FJOURNAL =. 1987 , NUMBER =. doi:10.1112/blms/19.5.481 , URL =

1987 doi
[239]

Cooper, Daryl and Long, Darren D and Reid, Alan W , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 1997 , NUMBER =. doi:10.1090/S0894-0347-97-00236-1 , URL =

1997 doi
[240]

2005 , PAGES =

Long, Darren and Reid, Alan W , TITLE =. 2005 , PAGES =

2005
[241]

Bass, Hyman and Lazard, Michel and Serre, Jean-Pierre , TITLE =. Bull. Amer. Math. Soc. , FJOURNAL =. 1964 , PAGES =. doi:10.1090/S0002-9904-1964-11107-1 , URL =

1964 doi
[242]

Weisfeiler, Boris , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1984 , NUMBER =. doi:10.2307/2006943 , URL =

1984 doi
[243]

Matthews, C. R. and Vaserstein, L. N. and Weisfeiler, B. , TITLE =. Proc. London Math. Soc. (3) , FJOURNAL =. 1984 , NUMBER =. doi:10.1112/plms/s3-48.3.514 , URL =

1984 doi
[244]

, TITLE =

Nori, Madhav V. , TITLE =. Invent. Math. , FJOURNAL =. 1987 , NUMBER =. doi:10.1007/BF01388909 , URL =

1987 doi
[245]

, TITLE =

Mennicke, Jens L. , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1965 , PAGES =. doi:10.2307/1970380 , URL =

1965 doi
[246]

Grigorchuk, Rostislav I , TITLE =. Izv. Akad. Nauk SSSR Ser. Mat. , FJOURNAL =. 1984 , NUMBER =

1984
[247]

Chong, Hip Kuen and Wise, Daniel T , TITLE =. J. Group Theory , FJOURNAL =. 2022 , NUMBER =. doi:10.1515/jgth-2021-0094 , URL =

2022 doi
[248]

Hyperbolic 3 -manifold groups that embed in compact Lie groups

Agol, Ian , year=. Hyperbolic 3 -manifold groups that embed in compact Lie groups
[249]

Virtually unipotent curves in some non- NPC graph manifolds

Douba, Sami , journal=. Virtually unipotent curves in some non- NPC graph manifolds
[250]

Anosov representations and dominated splittings , JOURNAL =

Bochi, Jairo and Potrie, Rafael and Sambarino, Andr\'. Anosov representations and dominated splittings , JOURNAL =. 2019 , NUMBER =. doi:10.4171/JEMS/905 , URL =

2019 doi
[251]

2001 , PAGES =

Kapovich, Michael , TITLE =. 2001 , PAGES =

2001
[252]

arXiv preprint arXiv:2206.03620 , year=

Special cubulation of strict hyperbolization , author=. arXiv preprint arXiv:2206.03620 , year=

arXiv
[253]

, TITLE =

Davis, Michael W. , TITLE =. Topology of high-dimensional manifolds,. 2002 , MRCLASS =

2002
[254]

Oh, Hee , TITLE =. J. Algebra , FJOURNAL =. 1998 , NUMBER =. doi:10.1006/jabr.1997.7188 , URL =

1998 doi
[255]

Benoist, Yves and Oh, Hee , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2010 , NUMBER =. doi:10.1093/imrn/rnp149 , URL =

2010 doi
[256]

Flag-no-square triangulations and

Przytycki, Piotr and \'. Flag-no-square triangulations and. Groups Geom. Dyn. , FJOURNAL =. 2009 , NUMBER =. doi:10.4171/GGD/66 , URL =

2009 doi
[257]

and Januszkiewicz, Tadeusz , TITLE =

Davis, Michael W. and Januszkiewicz, Tadeusz , TITLE =. J. Differential Geom. , FJOURNAL =. 1991 , NUMBER =

1991
[258]

arXiv preprint arXiv:2207.14356 , year=

Residually finite non linear hyperbolic groups , author=. arXiv preprint arXiv:2207.14356 , year=

arXiv
[259]

Michigan Math

Bestvina, Mladen , TITLE =. Michigan Math. J. , FJOURNAL =. 1996 , NUMBER =. doi:10.1307/mmj/1029005393 , URL =

1996 doi
[260]

Bestvina, Mladen and Feighn, Mark , TITLE =. J. Differential Geom. , FJOURNAL =. 1992 , NUMBER =

1992
[261]

Coxeter, Harold S M , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1934 , NUMBER =. doi:10.2307/1968753 , URL =

1934 doi
[262]

Geometry,

Vinberg, \`Ernest B and Shvartsman, Osip V , TITLE =. Geometry,. 1993 , MRCLASS =. doi:10.1007/978-3-662-02901-5\_2 , URL =

1993 doi
[263]

Internat

Stover, Matthew , TITLE =. Internat. J. Algebra Comput. , FJOURNAL =. 2007 , NUMBER =. doi:10.1142/S0218196707004165 , URL =

2007 doi
[264]

Proper CAT(0) actions of unipotent-free linear groups

Douba, Sami , journal=. Proper CAT(0) actions of unipotent-free linear groups
[265]

Anosov representations: Informal lecture notes , author=
[266]

Geometric and arithmetic properties of L\"obell polyhedra

Bogachev, Nikolay and Douba, Sami , journal=. Geometric and arithmetic properties of L\"obell polyhedra
[267]

Canary, Richard D and Stover, Matthew and Tsouvalas, Konstantinos , TITLE =. Bull. Lond. Math. Soc. , FJOURNAL =. 2019 , NUMBER =. doi:10.1112/blms.12248 , URL =

2019 doi
[268]

Gromov, Mikhail and Piatetski-Shapiro, Ilya , TITLE =. Inst. Hautes \'. 1988 , PAGES =

1988
[269]

Belolipetsky, Mikhail V and Thomson, Scott A , TITLE =. Algebr. Geom. Topol. , FJOURNAL =. 2011 , NUMBER =. doi:10.2140/agt.2011.11.1455 , URL =

2011 doi
[270]

arXiv preprint math/0612290 , year=

Systoles of hyperbolic 4-manifolds , author=. arXiv preprint math/0612290 , year=

arXiv
[271]

Geometry & Topology , volume=

Representations of polygons of finite groups , author=. Geometry & Topology , volume=. 2005 , publisher=

2005
[272]

Benoist, Yves , TITLE =. Geom. Dedicata , FJOURNAL =. 2006 , PAGES =. doi:10.1007/s10711-006-9066-z , URL =

2006 doi
[273]

Mathematische Annalen , volume=

Actions of discrete groups on nonpositively curved spaces , author=. Mathematische Annalen , volume=. 1996 , publisher=

1996
[274]

Mostow, G. D. , TITLE =. 1973 , PAGES =

1973
[275]

Publications Math

Quasi-conformal mappings in n -space and the rigidity of hyperbolic space forms , author=. Publications Math
[276]

2025 , howpublished =

Sami Douba and Franco Vargas Pallete , title =. 2025 , howpublished =

2025
[277]

Strong rigidity of

Prasad, Gopal , journal=. Strong rigidity of. 1973 , publisher=

1973
[278]

Three dimensional manifolds, Kleinian groups and hyperbolic geometry

Thurston, William P , journal=. Three dimensional manifolds, Kleinian groups and hyperbolic geometry. 1982 , publisher=

1982
[279]

1981 , PAGES =

Livne, Ron Aharon , TITLE =. 1981 , PAGES =

1981
[280]

The entropy formula for the Ricci flow and its geometric applications

Perelman, Grigori , journal=. The entropy formula for the Ricci flow and its geometric applications
[281]

Remarks on discrete subgroups with full limit sets in higher rank Lie groups

Subhadip Dey and Sebastian Hurtado , journal=. Remarks on discrete subgroups with full limit sets in higher rank Lie groups
[282]

Finite extinction time for the solutions to the Ricci flow on certain three-manifolds

Perelman, Grigori , journal=. Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
[283]

arXiv preprint math/0303109 , year=

Ricci flow with surgery on three-manifolds , author=. arXiv preprint math/0303109 , year=

arXiv
[284]

Proceedings of a

Homological group theory , SERIES =. Proceedings of a. 1979 , PAGES =

1979
[285]

Annals of Mathematics , pages=

On irreducible 3-manifolds which are sufficiently large , author=. Annals of Mathematics , pages=. 1968 , publisher=

1968
[286]

2025 , howpublished =

Minju Lee and Hee Oh , title =. 2025 , howpublished =

2025
[287]

Petri, Bram and Raimbault, Jean , title =. Comment. Math. Helv. , issn =. 2022 , language =. doi:10.4171/CMH/539 , keywords =

2022 doi
[288]

A series of word-hyperbolic

Felikson, Anna and Tumarkin, Pavel , journal=. A series of word-hyperbolic
[289]

There are no fake Seifert fibre spaces with infinite _1

Scott, Peter , journal=. There are no fake Seifert fibre spaces with infinite _1. 1983 , publisher=

1983
[290]

, TITLE =

Canary, Richard D. , TITLE =. Geometry, analysis and topology of discrete groups , SERIES =. 2008 , ISBN =

2008
[291]

Scott, Peter , TITLE =. J. London Math. Soc. (2) , FJOURNAL =. 1978 , NUMBER =. doi:10.1112/jlms/s2-17.3.555 , URL =

1978 doi
[292]

Journal of the London Mathematical Society , volume=

Compact submanifolds of 3-manifolds , author=. Journal of the London Mathematical Society , volume=. 1973 , publisher=

1973
[293]

Journal of Pure and Applied Algebra , volume=

A 3-manifold group which is not four dimensional linear , author=. Journal of Pure and Applied Algebra , volume=. 2014 , publisher=

2014
[294]

arXiv preprint arXiv:1703.05553 , year=

Properties of linear groups with restricted unipotent elements , author=. arXiv preprint arXiv:1703.05553 , year=

arXiv
[295]

McReynolds, David B , TITLE =. Algebr. Geom. Topol. , FJOURNAL =. 2004 , PAGES =. doi:10.2140/agt.2004.4.721 , URL =

2004 doi
[296]

Stallings, John , TITLE =. Invent. Math. , FJOURNAL =. 1983 , NUMBER =. doi:10.1007/BF02095993 , URL =

1983 doi
[297]

Coherence of 3-manifold fundamental groups , author=. S. 1977 , publisher=

1977
[298]

Proceedings of Georgia Topology Conference, Part 2 , year=

Problems in low-dimensional topology , author=. Proceedings of Georgia Topology Conference, Part 2 , year=
[299]

Auslander, Louis , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1967 , PAGES =. doi:10.2307/1970362 , URL =

1967 doi
[300]

Matematicheskii Sbornik , volume=

On isomorphic matrix representations of infinite groups , author=. Matematicheskii Sbornik , volume=. 1940 , publisher=

1940

Pith tools

Reviewed July 11, 2026 · model on record in the stance chip above.

[1] [1]

Journal of Topology , volume=

Virtual cubulation of nonpositively curved graph manifolds , author=. Journal of Topology , volume=. 2013 , publisher=

2013

[2] [2]

Ergodic Theory Dynam

Rees, Mary , TITLE =. Ergodic Theory Dynam. Systems , FJOURNAL =. 1981 , NUMBER =. doi:10.1017/s0143385700001206 , URL =

work page doi:10.1017/s0143385700001206 1981

[3] [3]

Geometry and dynamics of groups and spaces , SERIES =

Kapovich, Michael , TITLE =. Geometry and dynamics of groups and spaces , SERIES =. 2008 , MRCLASS =. doi:10.1007/978-3-7643-8608-5\_13 , URL =

doi:10.1007/978-3-7643-8608-5 2008

[4] [4]

Discrete subgroups with finite Bowen-Margulis-Sullivan measure in higher rank

Fraczyk, Mikolaj and Lee, Minju , journal=. Discrete subgroups with finite Bowen-Margulis-Sullivan measure in higher rank

[5] [5]

Aleksandrov, A. D. , TITLE =. Trudy Mat. Inst. Steklov. , VOLUME =. 1951 , PAGES =

1951

[6] [6]

Semisimple

de Cornulier, Yves , journal=. Semisimple

[7] [7]

Non-linear residually finite groups , JOURNAL =

Dru. Non-linear residually finite groups , JOURNAL =. 2005 , NUMBER =. doi:10.1016/j.jalgebra.2004.06.025 , URL =

work page doi:10.1016/j.jalgebra.2004.06.025 2005

[8] [8]

Convex cocompactness for

Danciger, Jeffrey and Gu. Convex cocompactness for. arXiv preprint arXiv:2102.02757 , year=

arXiv